The Dáil (Irish parliament) is still without a new leader after TDs failed to elect a taoiseach (prime minister).
The Independent Alliance, five rural independents, the Green Party, and TDs Katherine Zappone and Maureen O'Sullivan will take part in the talks.
Independent Thomas Pringle declined the offer.
The talks come after five rural TDs held what they called a constructive meeting with acting Taosieach Enda Kenny at Dublin's Government Buildings on Wednesday.
But one of their members, Denis McNaughton, said they had not yet made a decision on whether they would support any nominations for taoiseach.
A second vote on nominations for taoiseach will take place on 6 April.
Speaking on RTÉ's Morning Ireland, Mr Naughten said Fine Gael had momentum because it has extra seats.
He said he believes Fine Gael is more serious than Fianna Fáil about forming a government.
Mr Naughten said the group was not closing the door with Fianna Fáil, but that they wanted to move the issue along.
He said the group of independents were trying to see if they could come to some common ground on issues such as overcrowding in hospitals and the homelessness crisis.
He said the "merry-go-round" has gone on long enough and that it is time for people to get on with real talks.
Fine Gael will meet independent TDs (members of the Irish parliament) and others on Thursday for talks about the formation of a new government.