According to The Sun's Neil Ashton, United are hoping to sign Fred from Shakhtar Donetsk this summer but also admire Grealish as a player who could become a fixture in their side for years to come.
In particular, they are said to enjoy Grealish's "driving runs through midfield from deep-lying positions."
The 22-year-old helped Villa reach the play-off final this season, but he could be set to move on after the club failed to secure promotion against Fulham on Saturday.
An issue with his kidneys saw him miss the early part of the campaign, but the attacking midfielder still contributed three goals and five assists in 27 Championship matches.
Bleacher Report's Sam Tighe has been impressed with the youngster's development:
WhoScored.com's Martin Laurence praised him after the play-off final and hinted his future is now in question:
Martin Laurence @martinlaurence7
#AVFC fans absolutely sensational. Players gave everything second half and nearly had an 'I was there' moment from Grealish which would have been one of the best goals Wembley has seen in many years. Best player on the pitch. Best player in the Championship. What does he do now?
Despite his age, Grealish is capable of getting on the ball and dominating games from midfield, driving the team forward with technique and intensity, and creating chances for his team-mates.
That he is often targeted by opposition defenders with robust challenges is a testament to the threat he poses. His defensive work rate has also improved over the last year, which has no doubt further endeared him to United.
Broadcaster Ian Cheeseman believes Grealish possesses what might generously be described as uncompromising tenacity:
Red Devils boss Jose Mourinho typically welcomes such traits in his squad members. Grealish likely wouldn't get as much game time there next season as he would at Villa Park or at a lesser Premier League club, though.