Coveney in 'surprise' bid to battle Flanagan for Foreign Affairs post
Mr Coveney is understood to have expressed an interest in the job during his meeting with the incoming Taoiseach, in the aftermath of the Fine Gael leadership contest.
However, the Irish Independent understands that Mr Flanagan, who has held the post since 2014, has met with Mr Varadkar in recent days to make a pitch for why he should not be moved.
Mr Flanagan is understood to have briefed the new Fine Gael leader on his work in Northern Ireland and the relationship-building that has taken place with other European ministers since last year's Brexit vote.
Tánaiste Frances Fitzgerald is also understood to be interested in the role, but is seen as an outsider.
The job is set to create a major headache for Mr Varadkar, who is due to announce his new Cabinet on Wednesday.
Ministers have expressed "surprise" at Mr Coveney's bid for Foreign Affairs, given the political capital he has invested in trying to solve the housing and homelessness crisis over the past year.
However, one noted that it might suit Mr Varadkar to give the busy post to his leadership rival, as it will regularly keep Mr Coveney out of the country.
That will be a big call for the soon-to-be-former Social Protection Minister against a backdrop of political turmoil in the UK.
Although Mr Coveney appeared all but guaranteed whatever ministry he chose, his decision to target Foreign Affairs is problematic in light of recent events.
"Anybody who thinks they can just parachute into Northern Ireland talks or deal with other foreign ministers right now is optimistic at best," said one minister.
Another source said the DUP and Sinn Féin would "have a field day" and "throw curveballs" to test a new minister.
As well as Foreign Affairs, Mr Flanagan has been linked to a new department which will deal with home security.
Mr Varadkar's Cabinet will have a heavy focus on Brexit, with the post of European Affairs Minister likely to be elevated to 'super junior' status.
Dara Murphy, who currently holds that portfolio but backed Mr Coveney, is likely to be demoted.
Eoghan Murphy, who was Mr Varadkar's campaign manager, is believed to be in line for a central Brexit role in the Department of Finance.
It comes as Mr Varadkar's path to the Taoiseach's office was cleared by the Independent Alliance. The Alliance's five TDs confirmed yesterday they will support his election after Enda Kenny steps down.
The group, led by Transport Minister Shane Ross, met with the new Fine Gael leader last week but wanted issues clarified before giving him their backing.
At a second meeting yesterday, Mr Varadkar gave assurances that there will be more free votes in the Dáil and agreed a list of priority legislation with the Alliance.