The 44-year-old former Italy forward was sacked by Inter Milan 12 months ago, despite leading the club to three successive Serie A titles.
He has been out of football since leaving the San Siro club.
Mancini told Sky Italia: "I am honoured to have had an offer from Sunderland Football Club. It is a fantastic opportunity.
"It will be truly wonderful to lead such a prestigious and historic club in England after a year of sitting on the sidelines.
"I feel I am ready to start again with immense enthusiasm."
Mancini appeared to be an outsider, with Steve Bruce emerging as the firm favourite for the Stadium of Light post soon after Ricky Sbragia's resignation seven days ago.
Wigan have confirmed they will not stand in the way of Bruce, their manager, providing Sunderland pay £3 million in compensation.
Wigan chairman Dave Whelan has already given Bruce permission to talk to Sunderland.
Compensation, though, would not be a factor if Sunderland were to appoint Mancini.
Bruce joined Wigan from Birmingham in Nov 2007, and has been a success at the JJB Stadium.
Mancini has only limited experience of English football, having had a brief playing stint with Leicester in 2001 before going into full-time coaching.
Before joining Leicester he had a spell as player/assistant manager with Lazio, under Sven-Goran Eriksson, and later had spells as manager of Fiorentina and Lazio prior to joining Inter in 2004.