United are currently eighth in the Premier League table and six points behind fourth-placed Chelsea - who they face on Monday night.
They have been linked with a move for Grealish, who has impressed in the top flight this season after gaining promotion from the Championship with Villa.
While Lescott thinks the 24-year-old should consider leaving his boyhood club, he doesn't think Old Trafford is the right move.
"If he leaves Aston Villa he's going to want to play Champions League and that's not guaranteed at Man United, even in the future," the former Villans defender, who won two league titles with Manchester City, was quoted by GrandNational.org.uk.
"I would love to see him test himself at the highest level because I think he's capable of playing at real heights. He's proven that he can really perform in a team that's been struggling this season.
"In terms of if he was going to move to United, it would obviously be a great move for him because it's Man United. But in terms of how he can test himself against the very best, is he going to be able to do that at Man United as they're not guaranteed Champions League football?"
Despite Villa's struggles near the bottom of the table, Grealish has stood out with nine goals and seven assists in all competitions so far in 2019/20.
Smith believes most top sides should be interested in Grealish, with Tottenham Hotspur having previously made attempts to sign him.
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