Sinn Fein's single TD, the former bank official Caoimhghin O'Caolain, elected in Cavan-Monaghan, will almost certainly support Mr Ahern on the advice of its ard chomhairle (executive). Mr Ahern, however, has repeatedly said he will not seek that vote.
Mr Ahern's main priority is to draw in three independents with links to his party. Harry Blaney, elected in Donegal, is the brother of the late Neil Blaney, a former Fianna Fail minister who left the party during the 1970 Arms Trial crisis and took a hardline nationalist position on Northern Ireland. He continued to support Fianna Fail leaders in the Dail.
But Harry Blaney yesterday ruled out supporting Mr Ahern unless Fianna Fail made a commitment to seek Britain's withdrawal from Northern Ireland within a fixed time frame and promised a new referendum to outlaw abortion in Ireland.
Mildred Fox, who again held the Wicklow seat which was left by her late father who split from Fianna Fail after a selection row, is also considered sympathetic. Some believe she may join the Fianna Fail parliamentary party after consulting her supporters.
Another ex-FF member is the extrovert publican and Independent Jackie Healy-Rae, elected in South Kerry.
He is promising to "bang tables up in Dublin" in order to secure more public investment for his area, but will probably be behind Mr Ahern when the Dail meets on 26 June to elect a Taoiseach.
Mr Ahern has stated that he will not "do deals" with independents in order to secure their votes, but will consult those close to his party this week.
That approach was determined by the 1982 controversy when Charles Haughey won power by wooing the vote of the inner-city Dublin independent TD Tony Gregory through a deal costing an estimated pounds 70m for local redevelopment. Mr Gregory's remains another possible vote for Mr Ahern.
Sinn Fein increased its national vote by 0.9 per cent to 2.9 per cent, but as it ran only 15 candidates across 41 constituencies, this disguised larger support in targeted areas, notably where it was prominent in campaigns against alleged drug dealers.
In Kerry North the former IRA gun-runner Martin Ferris came close to taking a second seat, while Sinn Fein candidates also polled strongly in Dublin South-West and Louth.
The existing "Rainbow" coalition led by John Bruton will remain in office until a new cabinet receive their seals of office from President Mary Robinson after the Taoiseach's post is filled.
Mr Bruton may receive support in that vote from the two Greens and from Michael Lowry who resigned from his Fine Gael party last November.
The Democratic Unionist Party leader, the Rev Ian Paisley, said he did not think Unionists could do business with Mr Ahern's government because of Fianna Fail's continued support for Articles 2 and 3 of the Irish Constitution, which claim Northern Ireland as part of the Republic.
He said the people of Northern Ireland people be "enraged" by Mr Ahern's weekend comment that he would have one final meeting with the Sinn Fein President before becoming Taoiseach to emphasise the need for an IRA ceasefire.
Mr Ahern has said that once he is elected prime minister he will adopt the outgoing coalition's policy of refusing such direct contacts until the IRA formally abandons violence.
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