There are fears in the Green Party over convincing members to support a programme for government deal if they are in coalition with controversial Independent TDs.
Fianna Fail and Fine Gael are continuing to negotiate with Independents including Michael Lowry, Noel Grealish and Verona Murphy while still involved in talks with the Green Party.
It emerged over the weekend that Tanaiste Simon Coveney had called TDs from the Regional Technical Group, which includes Mr Lowry, Mr Grealish and Ms Murphy.
A senior Green Party figure said: "You would wonder what Fine Gael are playing at." They added that "members will have a problem with Lowry".
Another Green Party source noted the support of some Independent TDs, including Mr Lowry, was not the cause of undue friction during 2007-2011 coalition with Fianna Fail.
At that stage the Green Party concluded it could not dictate what other alliances Fianna Fail might enter into.
"But this time it could add to the difficulty of selling any coalition deal to the party membership. The two-thirds majority required by rule will be hard to achieve," the source added.
Mr Lowry did not respond to requests for comment.
The Tipperary TD has been convicted of tax offences and was criticised by the Moriarty Tribunal over his time as communications minister and the awarding of a mobile phone licence to Denis O'Brien-owned Esat Digifone.
He has voted with the Government on most issues over the last nine years and is a former Fine Gael TD.
Ms Murphy made controversial remarks about migrants and was deselected by Fine Gael before the General Election.
Mr Grealish caused public anger after he made Dail comments about Nigerians living in Ireland and sending money to their home country.
Meanwhile, the Green Party has dismissed suggestions it would seek to appoint a barrister with pro-life views as Attorney General.
In a statement, the party said "no names have been put forward by the party for the role of Attorney General in any potential coalition government".
"The party has not discussed possible appointments either internally or with other parties at any level," it added.
The statement was widely shared by Green Party TDs on social media, including deputy leader Catherine Martin.
Ms Martin shared the statement along with a photo of herself campaigning for a 'Yes' vote in the referendum to legalise abortion.
The statement followed a suggestion in the Sunday Times that barrister Roderick Maguire was a "likely candidate" for Attorney General should the Greens enter government.
Mr Maguire was among a group of 200 lawyers who signed a letter calling for a 'No' vote.
He was also reportedly a supporter of Green Party General Election candidate David Healy who also voted 'No'.
The three party leaders will decide on the make-up of the next Cabinet. The decision will be taken once a programme for government is agreed.
Work has progressed on developing a new super department which will have responsibility for housing and transport and function like the Department of Finance and Department of Public Expenditure and Reform.
Fine Gael have put forward the idea and senior Fianna Fail sources have suggested they will back the proposal.