Tuanzebe, 21, has enjoyed an encouraging loan spell in the Championship in 2018-19, making 26 appearances for Dean Smith's side as they made a late season surge to make the promotion playoffs.
With Villa holding an advantage over West Brom in the play-off semi-final after a 2-1 victory in the first leg, the defender could help the club secure a return to the Premier League.
However, Tuanzebe, who has five Premier League appearances for the Red Devils in his career, may well find himself in United's plans next season regardless of the sucess of Villa's promotion push.
The Red Devils struggled in defence in a frustrating 2018-19 campaign and Solskjaer is eager to see what Tuanzebe can do if given an opportunity in the pre-season.
"Hopefully, now Axel can get to the play-off final and play well," the United manager told the club's media channels.
"Obviously, he's had a couple of injuries, but he's a boy I can't wait to see in pre-season for us."
Solskjaer is also monitoring the progress of Henderson, who helped Sheffield United secure automatic promotion to the Premier League, picking up a Championship-best 21 clean sheets for the Blades as they finished second to Norwich City.
That success led to the keeper exclaiming his desire to return to Bramall Lane next season, and Solskjaer suggested the England Under-21 international could benefit from another spell with the South Yorkshire.
Henderson has, in successive years, gone from League Two, to League One and now the Championship and the United boss believes a year adjusting to the Premier League might be the best thing for him.
"Next year, let's see what we do with Dean," Solskjaer said.
"He's obviously won promotion with Sheffield United and that's been fantastic for him. If he plays a year in the Premier League for them, it might be the best thing."
While goalkeeper has been a position of strength in recent years thanks to the play of David de Gea, his end-of-year struggles, combined with his contract situation have created uncertainty around the future of the club's No.1.