The Independent Alliance says it is very optimistic to think a deal could be done by Thursday.
Shane Ross says it is up to Fine Gael to decide if they want to work with the group.
He said: "I think that if they agree to some of our most radical proposals and that we can say that Irish society has changed, then we will get an agreement.
"If, however, they baulk at it and they don't make concessions that are absolutely necessary for us, then I think it'll take an awful lot longer.
"It's really going to be up to Fine Gael to help this progress, it's not going to be any good if there is any obstruction or any delay or any tactics to resist change. I think it's up to them."
Daniel McConnell, Political Editor with the Irish Examiner, has been looking at the likely make up of the next Cabinet.
He said: "In order to get them on board, the Independent Alliance would be looking for two Cabinet seats and then you add Katherine Zappone who is pretty much guaranteed a Cabinet seat since she jumped first.
"And then the rural grouping will want at least one Cabinet seat."
Fine Gael is due to meet with independents today as the party works towards forming a minority Government.
It is the first set of meetings due to take place since the party agreed a deal with Fianna Fáil on Friday.
First up the Independent Alliance will meet with negotiators this morning, with the issue of a cardiac care centre for the South East believed to be the major stumbling block.
Fine Gael needs the support of six more TDs in order to elect Enda Kenny Taoiseach on Wednesday.
Mattie McGrath of the so called Rural Alliance says there is still more work to be done, particularly on paying for upgrades to the water infrastructure.
He said: "Going forward, we cannot have a situation where there are issues in our water services.
"I think there are many, many other days which I know they were discussing (water charges) as well but they didn't get the same prominence, they didn't dominate the discussions and we have to have our issues discussed as well, and resolved if we are going to make any progress."
The Minister for Agriculture and Defence Simon Coveney says there is a sense of urgency in forming the next Government.
Yesterday Independent TD from Waterford John Halligan said he would not be supporting a Fine Gael Minority Government, if they did not meet his demands for 24-hour cardiac care at Waterford hospital.
But at this stage Minister Coveney is optimistic that a deal could be done by Thursday.
Mr Coveney said: "There is asense of urgency, we've been working right through the Bank Holiday weekend on this issue, but I don't think it's fair us to share the documents that have been agreed between negotiating teams of Fine Gael and Fianna Fáil until the parliamentary parties of those two parties have approved those documents.
"Now once the negotitating teams conclude their discussions with the other, we will take those documents back to the parliamentary parties for final approval."