Mr Farrell was formerly secretary general of Fianna Fáil for much of the 1990s and served as a senator over that period.
He has been head of the IBF since 2004 and has played a prominent role in the response to the banking crisis.
He has helped shape the regulatory agenda and defended the banks against charges of squeezing the SME sector of credit.
Mr Farrell was previously head of marketing and communications at EBS Building Society between 1998 and 2003.
He has also held a number of positions in health administration, including a period as chief executive of Galvia Hospital in Galway.
Mr Farrell is a fellow of the Institute of Bankers. He is expected to take up the role in the next few months.
Dan Loughrey, who is retiring, has been at Bank of Ireland since Mar 2004. He joined the company from Aer Lingus.
Bank of Ireland is 15% owned by the State.