Prime Minister David Cameron's aides have privately sounded out government officials about Taoiseach Enda Kenny's interest in taking up a top EU post.
A cabinet minister confirmed for the first time yesterday that Mr Kenny is being linked with the appointments of the head of the European Commission and European Council.
The success of Ireland's EU presidency has put Mr Kenny at the centre of early speculation about the positions coming up for grabs next year.
Mr Cameron is believed to rate Mr Kenny highly and Downing Street feels the Taoiseach would have a greater understanding of Britain's often fraught relationship with the EU than a continental leader.
"One of the people pushing this name out there is the man in Downing Street (Mr Cameron). We have been getting questions from people in that building (No 10 Downing Street). He (Mr Cameron) has a high regard for him (Mr Kenny). They (the British) would feel he (Mr Kenny) would be sensitive to their sensitivities on Europe," a government source close to Mr Kenny said.
Mr Cameron has promised to put Britain's EU membership to a vote in 2017, provided he is re-elected as prime minister.
Mr Kenny has already voiced his opposition to, and concerns about, Britain leaving the EU.
Although Mr Kenny has not been asked directly about a move, German Chancellor Angela Merkel is believed to have him on her shortlist for a post. Fine Gael is a member of the European People's Party, the same European grouping as Ms Merkel's CDU party.
The EPP is likely to get one of the two posts when they are up for grabs next year.
Mr Varadkar said Mr Kenny's name does come up in connection with the Brussels positions when he attends EU meetings.
"I'm not surprised that he is being seriously talked about, because he is and the names that come up when you go over to Brussels and Luxembourg are himself, Jyrki Katainen in Finland and a Belgian guy, I can't think, maybe Guy Verhofstadt, I can't remember.
"He (Mr Kenny) is very much, you know, in the top three or four that people talk about and that's, I suppose, down to his performance as president when he had the (EU) presidency and also what's at least perceived in Europe, if not in Ireland, of a success story here in Ireland. So he is definitely one of the ones who is being approached.
"Whether he would like to do it, I don't know. I would think he wants to finish the job here, quite frankly, but that's for him to answer," he said.
Mr Kenny and his leadership style are well known to many EU leaders, as he has been attending EPP meetings for over a decade.
He ticks a lot of boxes as someone who comes from a small country and has an understanding of countries in economic difficulties.