Update 7.30pm: Fianna Fáil leader Micheál Martin says he simply couldn't break a promise he made to the electorate.
"The best interests of the Irish people are not served by a government made up by Fianna Fáil and Fine Gael.
"We made this promise consistently in advance of the election, we made it very clear to the Irish people and those who voted for us that we are not going into government with Fine Gael and we are remaining consistent and true to that commitment."
Update 6.15pm: Acting Taoiseach, Enda Kenny, has said Fianna Fáil's rejection of Fine Gael's offer to form a government "is a serious mistake and one which was driven by narrow party interests".
Mr Kenny said: "I regret that Fianna Fáil has refused to serve in a partnership government including Fine Gael, Fianna Fáil and Independents.
"Yesterday evening, I offered Micheál Martin a full partnership government. It is an offer that reflects the way in which people voted in the general election and the respective mandates of both parties and independents.
"Today, Deputy Martin rejected that offer. I believe that this decision is a serious mistake and one which was driven by narrow party interests rather than the national interest.
Enda Kenny: "FF's decision is a serious mistake, driven by narrow party interests rather than the national interest. #iestaff— McConnellDaniel (@McConnellDaniel) April 7, 2016
"Ireland needs a stable and lasting government to meet the many national and international challenges facing the country. Fine Gael's preferred option of a full partnership is the best option for providing the necessary stability and it is very regrettable that Fianna Fáil has rejected this historic opportunity."
Update 5.20pm: The Fianna Fáil leader has told the acting Taoiseach he cannot go into Government with Fine Gael.
Micheál Martin has told Enda Kenny he may be in a position to support a Fine Gael led minority Government.
Mr Martin has also questioned Mr Kenny's motives in the coalition offer, saying: "I was told last evening at the end of the meeting that minority government was being taken off the table.
"You listen to Simon Coveney this morning and he's adamant that it's not taken off the table, so I would have my own concerns in terms of what was afoot yesterday, to be frank.
"I don't believe it was sincerely put and I think there was an element of choreography about it, but that's par for the course in terms of how the Taoiseach does his business."
Update 4pm:Talks between Enda Kenny and Micheál Martin have finished.
Mr Kenny met with Micheál Martin this afternoon for less than 15 minutes, during which the Fianna Fáil leader said his party would not be accepting an offer of partnership government.
Meeting between Enda Kenny and Micheal Martin lasted less than 15 minutes #iestaff— McConnellDaniel (@McConnellDaniel) April 7, 2016
A spokesman for Enda Kenny has said Fianna Fáil's rejection of a partnership government deal is a “serious missed opportunity” and a “mistake”.
At a meeting between both leaders last night Mr Kenny put the possibility of a government made up of Fine Gael, Fianna Fáil and Independents to Mr Martin.
However, Fianna Fáil members rejected this idea at a meeting of the parliamentary party meeting today.
A spokesman for the Taoiseach described the rejection as “extremely disappointing, unfortunate, a mistake”.
Spokesman for the Taoiseach describes as a "mistake" Fianna Fail's decision not to accept partnership government #iestaff— Elaine Loughlin (@Elaine_Loughlin) April 7, 2016
He added that the “historic genuine offer” of equal government had been ruled out “with serious haste”.
“The Taoiseach believes this is a serious missed opportunity to act in the national interest. That it is a mistake."
He said a Fianna Fáil minority government or any other minority government was not discussed.
“At no point in the meeting did Deputy Martin put it to the Taoiseach that he should withdraw his lack of support for Fianna Fáil for a minority government."
He added that no further meetings between the two leaders have been arranged at this point.
“The Taoiseach said that he is respecting the Fianna Fáil mandate by making that offer that the electoral outcome is not what people were expecting but that in order to reflect the reality of the electoral outcome that it remains the case and that the offer still stands that a Fine Gael, Independent, Fianna Fáil government can provide the sort of stability that the country needs to address a number of issues which are increasing in urgency,” the spokesman said.
Update 2.55pm: Fine Gael and Fianna Fáil are at complete odds about Enda Kenny's offer of a "partnership" government after separate behind-closed-doors meetings with their TDs and senators today.
While Fine Gael unanimously backed a motion supporting the plan at a private parliamentary party meeting, Fianna Fáil voted against the offer after a four-hour discussion which ended just before 3pm.
In a statement on behalf of her parliamentary party, Fine Gael's acting parliamentary party chair Catherine Byrne said TDs and senators "overwhelmingly supported a partnership government".
She said it is "the best way to provide a stable and lasting government to deal with the issues concerning people and the challenges facing the country".
Secretary of the parliamentary party, backbencher Helen McEntee added the "historic offer" which will represent "seismic change in the political landscape".
Fianna Fáil separately decided in today's four-hour behind-closed-doors meeting to reject the offer on Wednesday night by Mr Kenny for a "partnership" government.
While a small number of TDs - including John McGuinness - said earlier today they may be in favour of a deal under certain requirements, a senior TD said the decision was unanimous.
The TD, who is also part of Fianna Fáil's negotiating team, said the party will continue speaking with Independents.
Fianna Fáil leader Micheál Martin is expected to speak to the media this evening after meeting with Mr Kenny to formally reject his proposal later today.
Update 1.50pm: Former Fianna Fáil justice minister Dermot Ahern has said a 50:50 ministerial breakdown and rotating taoiseach role "would have to be" part of any deal with long-term rival Fine Gael, writes Fiachra Ó Cionnaith of the Irish Examiner.
The senior figure of Micheál Martin's party said while he is "not in favour" of any deal, the offers must be included in any attempt to form a mooted partnership government.
Speaking on RTE Radio's News at One programme this afternoon, Mr Ahern said he "personally wouldn't be in favour of a grand coalition because it would only replace the outgoing government".
He said any grand coalition "would be bad for democracy" as a small majority agreement "keeps the government on its toes".
However, if acting Taoiseach Enda Kenny's Wednesday night deal is accepted by Fianna Fáil - a position that seems at this stage highly unlikely - Mr Ahern said it must include a 50:50 ministerial breakdown and a rotating taoiseach role.
Noting Mr Kenny has already confirmed he will step down over the course of the next government, should he be part of it, Mr Ahern said the rotating taoiseach scenario could be included in this situation as "he's [Mr Kenny] like a second term US president, a bit of a spent docket, is he an obstacle in formation of the next government".
The former Fianna Fáil justice minister said he understood why Fine Gael is "rapturously" backing a partnership government as "they lost the election".
While saying "all options" must be considered because of the way "the cards have fallen", he suggested Fianna Fáil may still seek to back a Fine Gael-led minority government from the opposition benches.
However, he said it "wouldn't be an easy position as we saw with the Tallaght Strategy. Fine Gael took years to recover from that arrangement".
Meanwhile, speaking on the same programme, acting Fine Gael Justice Minister Frances Fitzgerald repeatedly said the "partnership" offer was "historic" and what the country needs.
However, she was unable to explain what exactly is involved in terms of ministerial positions, the number of Independents who will be asked to take part and whether a rotating taoiseach scenario is included in the potential deal.
"How could I reply about the structure of this government... we're not into the level of that detail, how could I possibly predict it at that point," she said after stressing:
"This would be an historic change, we should just register that."
Update 1.35pm: Fine Gael TD Simon Harris, acting Minister of State at the Department of Finance and the Taoiseach, Enda Kenny and Micheál Martin need to be given more time to continue talking.
Mr Harris said: "I think it would be extraordinarily helpful and constructive if at the end of their meeting today and the end of our Parliamentary Party meeting today, that the leaders of Fine Gael and Fianna Fáil could be given space and the opportunity to continue talking.
"I think it would be extremely regrettable if the conversation was to be suddenly ended or brought to a sudden halt. There is much to discuss."
Update 11am: Limerick Fianna Fáil TD Willie O'Dea (pictured below) has said that in his view the proposed partnership government between Fianna Fáil and Fine Gael "won't wash".
"I personally would be opposed to it…From the soundings I've taken, which may be wrong, I would anticipate the majority of my colleagues will oppose it also."
Party colleague Jackie Cahill said: "We're equal partners, so I would say everything has to be on the table and obviously if there's a rotating Taoiseach, it would be more attractive the Fianna Fáil membership."
Meanwhile, Cork North Central Fianna Fáil TD Billy Kelleher said: "This is a long way off, in view of the fact there are alternatives available."
Former Fianna Fáil TD Dermot Ahern said he believed the grand coaltion would be rejected by Fianna Fáil grassroots.
"There would be divided opinion. I get the strong feeling there is very much an anti feeling towards going into coalition with Fine Gael," he said.
Update 10.45am: Anti Austerity Alliance Deputy Paul Murphy has warned against a partnership government between Fianna fáil and Fine Gael.
"I don't want to see a Fianna Fáil-Fine Gael government elected," he said. "I think it would be very right wing, a very conservative government…For the first time they're reduced to less than 50% of the popular vote."
Acting Minister for Agriculture Simon Coveney has said Enda Kenny's offer of a partnership government to Micheál Martin is sincere, and not a political manoeuvre.
He also said he wanted to send a message to Fine Gael members that "this is something we're serious about and want to be generous towards", and to Independents that "we (Fine Gael) haven’t forgotten about you".
"This is not a political manoeuvre," he said this morning.
Simon Coveney could not tell us earlier who would be named Taoiseach, nor whether it would rotate between the two party leaders.
"There isn’t an answer to that right now," he said. "What we have asked Fianna Fáil to consider is the principle of working with us to put a partnership government together.
"If they agree to move ahead with that, both sides will put negotiating teams in place to decide how that would work, and the practicalities of it."
The two leaders met for 45 minutes yesterday evening. Mr Kenny made the proposal offer to Mr Martin of forming a Government made up of Fine Gael, Fianna Fáil and Independents.
Mr Kenny stressed the need for stable Government in the meeting which was described as "businesslike".
Mr Kenny made the offer with the unanimous approval of his ministers.
Mr Martin did not commit to agree, but said he would have to refer the offer to his parliamentary party which is to meet this morning.
Even if a deal were thrashed out, it would have to go to a special Fianna Fáil Ard Fheis and many members say it would not receive the required backing.