He is currently the managing director of retail banking for the Ulster Bank group.
Mr Donnan replaces Ellvena Graham, who is becoming a non-executive director of the bank.
She held the position as â€Žhead of Ulster Bank Northern Ireland since it was created just over two years ago.
Ulster Bank is the largest bank in Northern Ireland and the third largest in the Republic of Ireland.
Its chief executive Jim Brown said: "I would like to wish Ellvena well as she departs her executive role.
"She has always been a positive and energetic influence on the business with a solid understanding of what matters for our people and our business.
"She has also been a great support to me personally."
Mr Donnan has been with Ulster Bank for about 25 years.
Ulster Bank has appointed Richard Donnan as its new boss in Northern Ireland.