The England international has struggled for regular action this season, with Solskjaer selecting a number of players ahead of the 28-year-old.
Lingard has been linked with a return to West Ham United in January, having enjoyed a successful loan spell at the club in the second half of last season, and he recently hinted that he could move to the London Stadium.
The likes of Donny van de Beek and Anthony Martial are both said to be prepared to wait before making decisions on their futures, with the pair both named in the Man United XI on Tuesday night against Villarreal, with Michael Carrick now picking the side.
However, Lingard was not used in the Champions League clash, and according to the Daily Mail, the attacker is still desperate to leave despite Solskjaer's exit.
The report claims that the Man United academy product is determined to secure regular football in 2022, as he bids to make the England squad for next year's World Cup.
Lingard's contract at Old Trafford is due to expire next June, and the Red Devils are allegedly prepared to sell him for £10m during the January transfer window. body check tags ::