Sources told ESPN's Mark Ogden earlier this month that City boss Pep Guardiola had made Grealish a key target along with Harry Kane, and now City are ready to pay around £100 million ($139m) for the midfielder. The deal would be a British record if it goes through, eclipsing the £89.3m ($124m) Manchester United paid for Paul Pogba in 2016.
City have so far declined to comment, but sources said a fee has yet to be agreed between the Premier League champions and Villa.
Sources have told ESPN formal talks between the clubs are yet to take place but there is a belief at City a deal is achievable after Euro 2020. Guardiola and director of football Txiki Begiristain are admirers and interest in Grealish has been growing for more than a year.
Grealish, 25, is preparing for England's European Championship round-of-16 clash with Germany, and his move to City is expected to progress once his involvement in the tournament is over. The final is scheduled to be held at Wembley on July 11.
Grealish has made 213 appearances for Aston Villa, scoring 32 goals, since making his senior debut for his boyhood club in 2014. He has won nine caps for England and started the 1-0 win over Czech Republic on Tuesday.
Sources have told ESPN that City will retain their interest in Kane even if they get a deal for Grealish over the line. Kane still has three years left on his contract and is valued by Tottenham Hotspur chairman Daniel Levy at more than £150m ($208m).
The potential arrival of Grealish is likely to throw the futures of a number of City's forwards into doubt. Sources have told ESPN that Bernardo Silva is open to leaving the Etihad Stadium amid interest from a number of clubs in Spain and Italy.
Raheem Sterling has also been linked with a move after falling out of favour under Guardiola for much of the second half of last season.