Solskjaer splashed nearly £150 million to bring Harry Maguire, Aaron Wan-Bissaka and Daniel James to Old Trafford during the transfer window and he is planning to sign more domestic talent at the end of the season.
Borussia Dortmund forward Sancho and Leicester midfielder Maddison top the list of targets with Solskjaer keen to inject more goals into his squad after allowing Romelu Lukaku and Alexis Sanchez to join Inter Milan.
United have not been put off pursuing Sancho despite the 19-year-old agreeing a new long-term contract at Dortmund in August. They expect Leicester to value Maddison, 22, at between £70m and £80m.
United will look to strengthen their midfield and forward line next summer after Solskjaer admitted he has been left short of attacking options after the departures of both Lukaku and Sanchez.
Saturday's 1-0 win over Leicester on Saturday moved United up to fourth in the Premier League table.
They have the joint best defensive record in the league after five games but have been restricted to just eight goals themselves -- eight fewer than top scorers Manchester City.
Meanwhile, sources have told ESPN FC that goalkeeper David De Gea has penned a new deal at Old Trafford.
De Gea agreed terms on a six-year deal earlier in the summer and the formalities have now been completed to extend his stay at the club.