Who would be ideal? And who is realistic?
THE DREAM CHOICE: Colm Kelleher.
Kelleher is Ireland's top banking export. President of Morgan Stanley's investment banking and trading division, he earned $14.5m in 2013. He helped to shape Morgan Stanely into what Banker's Bible the International Financial Review named its 2014 bank of the year.
The Oxford-educated Cork man - one of nine siblings - is well-known on Wall Street. He famously conducted a business meeting while lying on his back due to a sports injury. He made last year's Sunday Independent Rich List, so he won't be taking the AIB job for the pay. What might motivate him more is returning billions to taxpayers.
THE SAFE HANDS
Brady is head of Citibank Europe, one of the financial behemoth's biggest banking vehicles with operations across Europe. He also handles Citi's other Irish businesses, covering some 2,500 staff.
Another chartered accountant, he has three decades of experience in banking and long-standing links in the Irish banking community as he is the former head of the Irish Banking Federation. Brady is 60 years old.
Slattery is an executive vice president at State Street and heads up its Irish operation. He knows retail banking inside out - he worked for the Central Bank for 23 years and was deputy head of banking supervision until he left in 1996.
He is outspoken and not afraid to criticise the Government, having publicly found flaws with everything from the NTMA to the country's tax regime.
But we know that the Department of Finance likes him, given that Michael Noonan nominated him to the Aer Lingus board.
WHO IS REALISTIC?
Bourke (49) joined AIB as its chief financial officer late in 2013. He moved from pension and investment group IFG where he spent 12 years, reaching CEO before jumping ship. Crucially he joined after the salary cap of €500,000 was introduced (taking a pay cut to do so, because company accounts show him on €813,000 at IFG in 2012).
So we already know he will work at AIB under the pay cap.
Bourke began his career at PwC in 1989 and was made a partner in the firm in 2000.
Byrne (45) is a less obvious choice than Bourke given that he spent a lot of his career at a utility company rather than a financial institution. He was ESB's commercial director from 1994 to 1998 and then rejoined as chief financial officer for another six year stint in 2003.
He moved to AIB in 2010 as chief financial officer and then, unusually, moved across to head up its retail banking unit. He now also heads up corporate banking.
Working in his favour is the fact that he's been at the bank for more than four years, weathering out the bailout and its restructuring.
Sunday Indo Business