A source from the centre-right party told the Irish Independent there was "pretty unanimous support" for Mr Martin's decision.
Taoiseach Leo Varadkar said on 12 February he expects to be the leader of the opposition in the next Irish parliament - which would mean Fianna Fail and Sinn Fein are the most likely coalition partners.
Mr Martin made note of Waterford Sinn Fein TD David Cullinane's shout of 'up the Ra (IRA)' at a party bash after last weekend's election success.
Mr Varadkar's party, which has 35 seats in the 160-seat parliament, has already ruled out a deal with Sinn Fein, which is on 37.
There are open divisions among Fianna Fail lawmakers over talking to Sinn Fein.
The head of Sinn Fein said on Thursday it would be "very, very tricky" to form a government that does not include one of Ireland's two large center-right parties, Fianna Fail and Fine Gael. "We gave the party leader license to speak to whoever he needs to speak to, with the exception of a Sinn Fein", senior Fianna Fail lawmaker Niall Collins told Reuters on Thursday.
"The party is fully behind the position", he added.
Fine Gael and Fianna Fail have long shunned Sinn Fein, citing policy differences and the party's historic links to the IRA, which fought British rule in Northern Ireland for decades in a conflict in which some 3,600 people were killed before a 1998 peace deal. The Labour Party, which also has six seats, said it plans to sit on the opposition benches.
Surveys showed that voters rejected the traditional parties over the key campaign issues of healthcare and the high cost and low availability of housing, won over by Sinn Fein's high-spending promises and a pledge to freeze residential rents.
"We have always said we would speak to Fianna Fail but I think that would be a hard process", Fine Gael deputy leader Simon Coveney told RTE.
McDonald is to address Sinn Fein's new TDs in Dublin on 13 February and will meet with the Social Democrats later about the possibility of government formation.
But she said a Fianna Fail-Fine Gael coalition would be "a slap in the face to the Irish electorate" who had clearly expressed a desire for change. He's got a mandate.
But Foreign Minister Simon Coveney of Fine Gael repeated his personal view that another minority government was not a good idea after both parties suffered in the election.
"There are a number of minority type administrations that could be put together with each of the three parties involved and essentially underscoring a confidence and supply arrangement", Collins told national broadcaster RTE.