Grealish, according to sources close to the midfielder, has been torn over his decision to leave his boyhood club this summer but there is a growing belief at the Etihad Stadium that the 25-year-old will be their player before the start of the new season.
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Pep Guardiola's side begin their Premier League campaign against Tottenham Hotspur on Aug. 15.
City are set to pay £100 million for Grealish to activate a release clause in his contract at Villa Park. Sources have told ESPN the idea of including highly rated 19-year-old winger Morgan Rogers in the deal has been floated, but Villa have indicated their preference is a straight cash bid.
Grealish reported to training with Villa on Sunday after a holiday with friends following Euro 2020.
Sources have told ESPN that the chance to play Champions League football and challenge for trophies at City is key to Grealish's willingness to leave Villa.
Villa are already working on bringing in reinforcements and on Saturday announced they have agreed a deal with Bayer Leverkusen for winger Leon Bailey.
Meanwhile, City's hopes of adding Harry Kane to Guardiola's squad have been boosted after the striker failed to return to training with Tottenham on Monday.
City have made Kane their top target to replace Sergio Aguero, but sources have told ESPN they will not meet Spurs' reported asking price of more than £150m.
City value Kane at closer to £100m, but there is an acceptance at the club a deal will be hard to get over the line given he has a contract at Tottenham until 2024.