Earlier today, Mr Martin and acting Taoiseach Enda Kenny both failed to be elected Taoiseach in the Dáil on their third attempt each.
The motion to elect Mr Kenny was defeated by 77 votes to 52, meaning he had gained just one additional vote on the previous occasion - Independent TD Katherine Zappone.
The party now says it will not be putting Mr Martin forward for a fourth time.
He has been invited by Enda Kenny to meet to see how the two parties continue with Government formation talks which the Acting Taoiseach says must now take on a "sense of urgency".
And, Mr Martin has signalled he's willing to negotiate.
Dublin South-West TD Ms Zappone was one of only two Independents who have voted for Mr Kenny, along with Tipperary TD Michael Lowry.
Ms Zappone is a new TD elected this year, having previously been appointed by Mr Kenny to the Seanad in 2011.
Yet, following Mr Kenny's defeat, Fianna Fáil leader Micheal Martin also lost his bid to become Taoiseach by 43 votes for, to to 91 against.
Mr Martin failed to attract any more Independents than Deputy Kenny.
Mr Martin had previously given Independent TDs a final ultimatum to support him in a minority government last night.
In nominating her party leader Micheál Martin, Fianna Fáil deputy Lisa Chambers told TDs that if they "sit on their hands or vote against" Micheál Martin in this afternoon's vote for Taosieach, it will effectively put Enda Kenny back in office.
It is the third time the Dail has gathered to vote a Taoiseach, some seven weeks after the election.
