Update 7.31pm: British Prime Minister Theresa May has urged Stormont's leaders to make one final push to strike a deal to salvage powersharing in Northern Ireland.
After holding talks with the Democratic Unionists and Sinn Féin in Belfast, Ms May insisted there was the basis of an agreement to end the 400-day political impasse and she expressed confidence a devolved executive would be "up and running very soon".
Taoiseach Leo Varadkar also held talks with politicians at Stormont, though notably he did not have a meeting with the DUP.
Mr Varadkar said he hoped a deal could be achieved later this week.
Mrs May, who met all the main parties at Stormont House this afternoon, said the discussions were "full and frank".
"The DUP and Sinn Féin have been working hard to close the remaining gaps," she said.
British PM says there is the basis for agreement to restore NI power-sharing pic.twitter.com/1lDZcWWXbQ— RTÉ News (@rtenews) February 12, 2018
"Some differences remain but I think there is the basis of an agreement here and I have been urging the parties to make one final push for the people of Northern Ireland."
Mr Varadkar said the differences were "not insurmountable". He expressed hope that Stormont's smaller parties - the Ulster Unionists, SDLP and Alliance - could be part of a new coalition executive.
"It is our strong view that an inclusive executive including as many parties as possible would be more sustainable and more beneficial for Northern Ireland as a whole," he said.
Relations have been strained between the DUP and Mr Varadkar's government over the Brexit process, but Ireland's deputy premier Simon Coveney urged people not to "read too much" into the fact that the main unionist party did not meet the Taoiseach at Stormont on Monday.
Northern Ireland has been without a functioning government since January 2017 and several rounds of talks to resolve the crisis have failed.
However, speculation has been growing that a deal between Sinn Féin and the DUP is imminent.
DUP leader Arlene Foster said the tone of the meeting with Mrs May was positive. Mrs Foster said she shared public frustration that the pace of progress had been slow.
DUP leader Arlene Foster says progress made in NI talks but they'll work to look for more progress pic.twitter.com/ntmAiwm48x— RTÉ News (@rtenews) February 12, 2018
"We have been engaged in three weeks of intensive talks with Sinn Fein building on the talks that have been going on for some time," she said.
"We recognise the frustration that is out there but good progress has been made and we will continue to work to look for more progress."
Sinn Fein President Mary Lou McDonald said: "We believe that we are close to an agreement which, certainly, we can put to our grassroots and to the community as a whole."
Sinn Féin leader Mary Lou McDonald says nothing insurmountable left to resolve in NI talks pic.twitter.com/7ZMAjwAOto— RTÉ News (@rtenews) February 12, 2018
She said "we are not exactly there just yet", but added: "there is nothing insurmountable if there is the political will to reach an agreement".
The DUP/Sinn Féin-led powersharing arrangement imploded last January amid a row over a botched green energy scheme.
That rift subsequently widened to take in long-running disputes over culture, social issues and legacy.
The main sticking point preventing the restoration of an executive is the Irish language.
Sinn Féin wants a standalone piece of legislation to protect speakers - an Irish Language Act - but the DUP has long insisted it would only countenance new laws if they also incorporate other cultures, such as Ulster Scots.
There was speculation over the weekend that three pieces of legislation - an Irish Language Act, an Ulster Scots Act and a broader Culture Act - could be a means to satisfy both sides.
Earlier: Northern Executive could be 'up and running very soon', says Theresa May
Update 6.15pm: British Prime Minister Theresa May said today that "while differences remain, I think there is the basis of an agreement here" and hoped that the Northern Executive could be "up and running very soon".
She urged Northern Ireland's political leaders to make "one final push" to reach an agreement.
Ms May and Taoiseach Leo Varadkar, as well as the leaders of Sinn Féin and the DUP, were involved in a series of talks today.
Earlier this evening, Mr Varadkar said there was a "lot of work to do" but progress was going in the "right direction".
"We are very hopeful that those two parties are able to come to an agreement this week," he said.
Democratic Unionist leader Arlene Foster said earlier that the tone of the talks had been "very good". Sinn Féin president Mary Lou McDonald said the two parties were "close to an agreement".
Earlier: Taoiseach 'hopeful' that new powersharing deal can be agreed
Update 6pm: Taoiseach Leo Varadkar has said that he is optimistic that a new powersharing deal can be reached after new talks concluded today.
"I think the soundings from both Arlene Foster and Mary Lou McDonald were positive and I'm hopeful that they can make an accommodation in the days ahead," he said.
"It doesn't necessarily require that the Prime Minister and I return for that, but in our role as co-guarantors of the Good Friday Agreement, we will, of course, support and try to facilitate any agreement they make."
When asked what he felt was different about this round of talks, Tánaiste Simon Coveney said: "I think there has been a lot of talking, and there has been, I hope, a process that has helped to rebuild relationships, and some trust,
"The DUP and Sinn Féin negotiating teams have spent a lot of time with each other, and indeed they've spent a lot of time with the two Governments as well,
"I think everybody has said for months now that they want to see devolved Government back up and running in Northern Ireland. It really is a very fundamental part of the agreement that keeps peace, so I think there is a determination to find accommodation for each other's positions, even though the positions at the outset were quite different.
He added: "We're not quite there yet, and the two parties can speak for themselves on that... but I think we are at a point now where both parties realise that if we're going to see devolved Government back up and running soon in Northern Ireland, well then parties need to make decisions. Hopefully, we can see that happening this week."
It is understood that British Prime Minister Theresa May will be speaking to the media later this evening.
Earlier: No deal yet, but 'good progress' on new powersharing deal
No agreement has been reached in efforts to broker a new powersharing deal in Northern Ireland today, despite upbeat comments from party leaders.
Speaking to the media at Stormont Parliament buildings this evening, DUP leader Arlene Foster said: "There isn't a deal yet. What there is is very good progress.
"And we're going to keep at it, and continue to work on that progress, and that's why we're here today."
Speaking shortly afterwards, new Sinn Féin leader Mary Lou McDonald confirmed that progress towards a new deal has been made.
This afternoon she met with British Prime Minister Theresa May and Taoiseach Leo Varadkar but has yet to meet with the DUP.
"We believe that we are close to an agreement which, certainly, we can put to our grassroots and to the community as a whole," she said.
"We're not exactly there just yet, but let me repeat - there is nothing insurmountable. If there is the political will to reach an agreement, we certainly have that determination and that resolve."
Ms May and Leo Varadkar are due to speak to the media shortly.
More to follow.