The Red Devils are expected to be active in the upcoming market despite the financial problems caused by the coronavirus outbreak, which has halted the majority of football across the globe.
Grealish has been in impressive form for Villa during the 2019-20 campaign, scoring nine times and contributing eight assists in 31 appearances in all competitions.
United have been consistently linked with the Englishman over the last few months, and according to The Sun, the 20-time English champions are now 'stepping up their pursuit'.
The report claims that Villa value the 24-year-old in the region of £70m, although the price could drop to £50m if Dean Smith's side - currently 19th in the Premier League table - are relegated this season.
Grealish's existing deal at Villa Park is due to run until the summer of 2023.