Speaking at the White House, following a meeting with US President Barack Obama, Mr Kenny said Fianna Fáil has responsibilities in seeing a stable administration being formed.
Last night, Fianna Fáil said it has always acted responsibly, raising hopes that formal talks between the two largest parties may take place.
Mr Kenny’s truncated trip to Washington for St Patrick’s Day celebrations were overshadowed by the political uncertainty with Vice-President Joe Biden pointedly referring to the Taoiseach as having “lost” the election.
Mr Obama warned of “volatility in the electoral process” after the recent general election during his final meeting Mr Kenny. However, Mr Kenny made his most explicit plea to Mr Martin to enter talks with his party in the coming days.
“I have issued an invitation from the Dáil already, that my responsibility is to work to provide a lasting and working government for the people and I have invited like-minded parties to work with us in that regard,” said Mr Kenny.
“I hear different stories from Fianna Fáil, when I have my priorities agreed and put in place, I have invited them already to work with us in the sense of putting together a government that will last and do its duty in respect of the people.”
Mr Kenny said Fine Gael expects to finalise its list of priorities which it would see as a basis of negotiations with other parties.
“So we expect to have that finalised next week and that gives us the basis for negotiations and discussions about putting a government together,” he said.
Mr Kenny said he had not yet spoken to Fianna Fáil.
Meanwhile, the Social Democrats last night ruled themselves out of any Fine Gael-Fianna Fáil alliance.
After meeting with Fianna Fáil yesterday evening Catherine Murphy said the party had come to the conclusion that the current exercise of Fianna Fáil and Fine Gael talking to smaller parties and Independents is “only postponing the inevitable”.
Speaking at a breakfast for members of the Irish-American community, hosted by Mr Biden, Mr Kenny promised that Ireland would see a stable government formed in the coming weeks.
In highly political remarks, Mr Kenny was highly critical of the anti-austerity parties.
Mr Kenny referenced the impact of hard-left groups in Europe which he said seek to run away from responsible government and do not have the courage to do what is right for their people.
In reference to the recent general election, Mr Biden paid tribute to Mr Kenny despite the fact “he lost” the election.
“He is the most popular guy in the State and he lost,” said Mr Biden.
“I can assure you, if you ran in America you would get 80% of the vote,” he said.
At an Irish Embassy event last night, Mr Kenny joked that he wished he didn’t have to go home.
“Bejaysus, I wish I didn’t have to go back and face what I have to face,” he said.
Mr Kenny described the verdict of the electorate as a complex one but he said he respected it.