Solskjaer is a fan of Grealish and urged executive vice-chairman Ed Woodward to enquire about his availability last summer and in January.
Both approaches were knocked back but there is belief Aston Villa may sanction a sale during the next transfer window, regardless of whether or not they stay in the Premier League this season.
Grealish has been in the news for the wrong reasons after it emerged he had broken UK government advice to stay at home during the coronavirus pandemic and attended a friend's party.
The 24-year-old has apologised for his actions and while the incident raised eyebrows at Old Trafford, Solskjaer is keen to continue working towards a deal.
The United boss is confident that the processes put in place by the recruitment department to find out more about a target's character and life away from football will answer any outstanding questions but he has already been impressed with the leadership Grealish has shown at Villa Park since becoming captain of his boyhood club.
Grealish is open to a move to United and is already a regular visitor to Manchester in his spare time having found the attention on him in Birmingham too intense. In March 2019, he was assaulted by a Birmingham City fan who ran on to the pitch during a game.
Villa are likely to demand upwards of £60 million for Grealish, who is under contract until 2023, although United chiefs are expecting transfer fees to drop across the board once football returns after the coronavirus shutdown.
United are also still interested in Leicester midfielder James Maddison but there is a feeling that Grealish is more attainable and it is unlikely the club will sanction moves for both in the same window.
Solskjaer has said that United should be in a position to "exploit" the transfer market once the window opens following the coronavirus shutdown, with sources telling ESPN that Harry Kane is on the club's transfer shortlist despite Tottenham's stance that they will not sell their star striker to a domestic rival.
United are also ready to rival Real Madrid for Erling Haaland if the striker leaves Borussia Dortmund, sources have told ESPN.