DUBLIN — Ireland remained mired in political limbo Wednesday after lawmakers overwhelmingly rejected all three candidates to lead the country’s next government.

It was the second time that lawmakers tried and failed to select a prime minister following Ireland’s Feb. 26 election. That poll 40 days ago left the two traditional enemies of political life — caretaker Prime Minister Enda Kenny’s Fine Gael and Micheal Martin’s Fianna Fáil — virtually neck and neck in a fractured Parliament. Their center-right parties trace their origins to the opposite sides of Ireland’s 1922-23 civil war and have never shared power.

But after both men failed to win the leadership vote, Kenny and Martin confirmed that they would meet Wednesday night to open discussions on forming a possible coalition. Kenny received all 51 votes from his own party, Martin 43 from his own party. A third candidate, socialist Ruth Pottinger, received 10 votes. A winner would have required at least 79 votes in Ireland’s 158-member Parliament.

A glum-faced Kenny, who has been Ireland’s leader since 2011, said he would try to form a government with Fianna Fáil that could survive a full five-year term.

‘‘I hope the discussions that I hope to initiate with Deputy Martin will lead us very much in that direction,’’ he said. Members of Irish Parliament are called deputies.

But Martin highlighted the possibility that Fianna Fáil could refuse to enter a coalition government and instead offer vote-by-vote support to a minority Fine Gael government. ‘‘The notion that the government must win every vote and get its way on every issue is a nonsense,’’ said Martin, who was foreign minister in Ireland’s previous Fianna Fáil-led government.

Should their coalition talks fail, Kenny could call a second 2016 election.