Richard Bruton and Simon Coveney linked with Health job
Leo Varadkar has been continually associated with the Health Minister's post. But Government sources say Taoiseach Enda Kenny needs a "safe pair of hands" for the role and Mr Bruton and Mr Coveney are being cited.
"You've got 18 months to steady the ship, to pick three things and do them. Sort out the medical cards, do some repackaging and get on top of the spending issue. It may not be Leo," a senior Coalition source said.
Dr Reilly is being linked with the Arts Minister's job or a junior ministry.
"As deputy leader, something is going to have to be found for him," a source said.
Mr Kenny needs to make health less high-profile, so Mr Varadkar, an independent-minded minister who could adopt a hardline in budget talks, may not suit.
"You put Leo into Health and he's too box office," a Coalition source said.
Ms Burton wants the junior coalition partners to get the Jobs Minister portfolio.
The Labour Party leadership favourite is willing to sacrifice Foreign Affairs in talks with the Taoiseach. She is keen to retain Social Protection and Education for Labour but the Communications brief is also on offer. However, there are concerns in Fine Gael over Labour having responsibility for union recognition.
The Coalition parties are still squabbling over who gets the plum post of European Commissioner, amid concerns the Government is moving too slowly and will get stuck with the "Commissioner for Paper Clips".
"Everything must be discussed in the round. They [Fine Gael] do covet it. It's definitely in the mix," a senior Labour source said.
Coalition sources are warning neither Phi Hogan nor Mr Gilmore may get the job, with Mairead McGuinness's name also being mentioned.
"The difficulty for Phil is his name has been on the table for so long. It could be someone else entirely," a Fine Gael source said.
Fine Gael figures are surprised by Mr Gilmore so openly canvassing for the job.
"Normally you don't ask to become a member of a club like that," a Government source said.
"He has got it to the stage now that it will be very embarrassing if he doesn't get it," a source said.
But Labour figures feel Mr Gilmore will help Ireland get a better portfolio.
"The thing about Gilmore, that would be in his favour, would be his reputation and he would be well known so he could get a good job," a Government source said.