The 29-year-old will see his contract expire at Derby County this summer, meaning he will be available on a free transfer.
Morrison made 36 Championship appearances for the Rams last season, scoring four goals and providing four assists.
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After coming through the ranks at Manchester United, Morrison moved to West Ham in 2012 and has struggled to settle at a club despite spells at Middlesbrough, Sheffield United and Cardiff.
Morrison said in March that he “definitely” wanted to sign a new contract at Derby.
Speaking back in March, before Derby were relegated to League One, Morrison was asked whether he’d like to stay at Derby.
Morrison told the Derby Telegraph: "Yes, definitely.
"It’s a massive club, the fans are great.
"I would love the team to stay up and we get a new owner, and we keep the team and push for something next year.
"We will have to see what the new owners do and what’s going on with the league.
"A lot of managers look at me and don’t want to give me the chance, but Wayne [Rooney] has put his trust in me, given me the chance, and I am enjoying my football and want to repay him.
"The manager has given me a chance and believed in me. It is nice when Wayne gives you a chance – he is like similar to me. Obviously he is a lot better, but he is similar to me in being a player who always wants to get on the ball.
“He knows what it is like and he helps me. It is enjoyable to play underneath him."
Sunderland are looking to strengthen ahead of the new Championship campaign, which will see Sunderland host Coventry City at the end of July on the opening weekend of the new campaign.
Sunderland won promotion via the League One play-offs.