City boss Pep Guardiola has identified the England international as one of his primary summer targets, along with Tottenham Hotspur star Harry Kane.
The 35-year-old is expected to leave Villa Park at the end of the season when his contract with the Midlands club expire next month.
According to the Daily Telegraph, Man United have already approached the veteran goalie, who is keen to return to Old Trafford, where he started his career in the youth academy.
The Reds and Spurs are being linked with a potential summer bid for the England Under-21 international, who has two years left to run on his current contract at Villa Park.
Jack Grealish is enjoying the best season of his life. Aston Villa captain seems to be living his dreams these days. He is the main man, the talisman of his boyhood team, the one he adores. He managed to leave the team in the top flight of English football with a stunning display on the last day of the season in the previous campaign.
Non league clubs have made it known on social media that they are interested in signing former Aston Villa player Gabby Agbonlahor.
The retired striker, who played 11 seasons in the Premier League and four in the Championship, could be about to return to playing again.
He took to his Twitter account, tweeting that he is tempted to come out of retirement at the age of 34, the play for the love of the game.