Cameron presidency threat puts Kenny in frame for Brussels job
It has been reported in international press that Mr Cameron would no longer be able to guarantee that Britain would remain a member of the European Union if European leaders elect Jean-Claude Juncker as EU Commission president.
French president Francois Hollande wants a French person to take the position and is also trying to block Mr Juncker's election.
Mr Kenny is seen as a "compromise candidate" for the €300,000-a-year job and would have the backing of many EU leaders should he be interested in putting himself forward.
Jose Manuel Barroso is set to step down this summer after 10 years in the job.
The commission president is selected by EU leaders but must be approved by the assembly, where Eurosceptics from the right made gains in last week's election.
The European People's Party (EPP), which won the most seats in the vote, had chosen Luxembourg's former premier Mr Juncker as its candidate at its March conference in Dublin's convention centre.
Although he too has been keen to support Mr Juncker publicly, Mr Kenny is seen as "an energetic 63-year-old", with Reuters reporting that he is popular among other EU leaders.
The only negatives mentioned in connection with Mr Kenny are his lack of fluent French and his deep roots in Irish rather than European politics which may undermine his candidacy.
German newspaper 'Der Spiegel' reported Mr Cameron said Mr Juncker was not an acceptable choice for Britain.
The magazine said participants understood Mr Cameron's comments to mean that a majority vote for Mr Juncker could destabilise his government to the extent that an "in-out" referendum would have to be brought forward.
'Der Spiegel' said Mr Cameron, who regards Mr Juncker as too federalist and likely to damage his hopes of reforming Britain's EU ties, dismissed the candidate during a recess with the words: "A face from the '80s can't solve the problems of the next five years."
That in turn, they understood, would most likely lead to the British people voting to quit the EU, the magazine said.
Given that vocal opposition to Mr Juncker, Mr Kenny's name has re-emerged.
Another frontrunner is Polish premier Donald Tusk, who has become an influential voice in EU talks in part because of Poland's economic strength and 38 million people.
Beyond the sitting centre-right prime ministers, there are technocrats who might also emerge in the coming weeks of negotiations, including Christine Lagarde, the current head of the International Monetary Fund.