Controversial TDs Michael Lowry and Verona Murphy have said they are open to supporting a coalition government following a meeting with the leaders of Fianna Fail, Fine Gael and the Greens.
Mr Lowry, Ms Murphy and six other TDs from the Rural Independent Group held a "productive" meeting with the leaders of the three parties attempting to form a coalition for more than an hour in Government Buildings on Wednesday.
Leo Varadkar, Micheal Martin and Eamon Ryan told the group they hoped to finalise a programme for government early next week and would present it to them.
They told the TDs it was up to them as to what extent they wanted to support a future coalition, saying they could strike a deal as a group or individually.
The group's other members are Cathal Berry, Sean Canney, Peter Fitzpatrick, Noel Grealish, Denis Naughten and Matt Shanahan.
"We said we wanted to be positive in relation to the government formation and a programme for government and we know the necessity of doing it," Mr Lowry said afterwards.
Those in the meeting said they saw a "good rapport" between the three leaders.
Mr Lowry is a former cabinet minister who left Fine Gael in acrimonious circumstances in the 1990s. He was found guilty of tax offences in 2018.
Ms Murphy fell out with Fine Gael over comments about migrants when she ran unsuccessfully for the party in the Wexford by-election last year.
Ms Murphy, who was elected as an Independent TD in February, said there were no issues between her and Mr Varadkar.
The Taoiseach was heavily critical of her last year, saying it was a mistake for Fine Gael to pick her as a candidate and that he was glad she was not elected.
However, Ms Murphy said: "The body language was very good, nobody has a problem with anybody.
"I got the impression that they would all like us to support it, but they are under no illusions - we need to see the programme for government."
One Independent TD said he believed it was an open-ended offer from the party leaders which could mean some TDs joining the next coalition.
"It's our decision how deep a relationship we want to have with them, rather than theirs," they said.
However, another TD said: "Whatever support the Government is going to get is going to be from the outside.
"I don't think there are any ministries up for offer."