The Premier League champions made the nine-figure offer for the 25-year-old last week, which would represent a British transfer record should Villa accept the bid.
According to football journalist Fabrizio Romano, negotiations are ongoing between both parties and these talks will be key if an agreement is to be reached this week.
If a deal for Grealish to join City fails to materialise, then the midfielder is likely to sign a new and improved contract to stay at Villa Park.
The England international has been identified by Pep Guardiola as one of his main targets this summer, as the Spaniard looks to build a stronger squad to try and retain their top-flight title as well as challenge for the Champions League.
Grealish, who returned to Villa's Bodymoor Heath training ground earlier this week, made 27 club appearances across all competitions last season, scoring seven goals and providing 12 assists. body check tags ::