Bernard Byrne, director of retail and business banking at First Trust parent Allied Irish Banks (AIB), is leading the race to replace David Duffy as the bank's chief executive, sources inside the bank say.
The bank has a shortlist and a second round of interviews for the post is set to be held within the next two weeks, the Republic's Finance Minister Michael Noonan said.
Mr Noonan said the list includes current AIB employees as well as external candidates.
"They were organising two rounds of interviews," Mr Noonan said. "The first was kind of informal, having a chat, and then they were having formal interviews.
"I'd say they are this week or next week, the next couple of weeks."
Other internal candidates in the running for the post are thought to include chief financial officer Mark Bourke and director of corporate and institutional banking Fergus Murphy.
AIB chairman Richard Pym has said the bank needed to "get a move on" as it hunts for a replacement for Mr Duffy.
Mr Duffy is leaving in the summer to take over at Clydesdale Bank in Scotland.
The new chief executive's salary will be capped at €500,000.
Mr Pym said he didn't think the cap would hinder the bank's efforts to find a good chief executive.
AIB is currently trying to rearrange its complicated capital structure before a return to private ownership.
At an event in Dublin last week, Mr Noonan said the earliest possible date for beginning to sell the Irish state's near 100% stake in the bank is October of this year.
He said the Government would explore floating 25% of the bank on the London Stock Exchange, and strongly emphasised that the timing of any sale won't be influenced by the next election.
Mr Pym said AIB will be ready for a sale whenever the Government is.