Lingard is desperate to find regular first-team football after making just three appearances so far this season. Solskjaer is sympathetic to the 28-year-old's situation but is keen to keep him at Old Trafford this month as he prepares for a busy second half of the season with games in three competitions.
United have exercised a one-year extension in Lingard's contract that will keep him at the club until 2022 but he remains keen to play regularly after falling down the pecking order under Solskjaer.
He impressed from the start in the 1-0 win over Watford in the FA Cup third round but was left out of the squad for the following two games against Burnley and Liverpool. His only other start this season came in the 3-0 win over Luton in the Carabao Cup in September.
Lingard, capped 24 times by England and a starter at the 2018 World Cup, has plenty of suitors in the Premier League and abroad. Marseille, Porto, Inter Milan, Nice, Tottenham Hotspur, Sheffield United and West Ham United have all registered an interest but United have not yet provided a definitive answer as to whether the former academy graduate is available.
Solskjaer has already allowed Timothy Fosu-Mensah to join Bayer Leverkusen and given Marcos Rojo and Sergio Romero permission to speak to other clubs -- and the Norwegian is conscious of losing too many squad players in January ahead of a hectic schedule in the Premier League, FA Cup and Europa League between now and May.
United could play as many as 34 games before the end of season if they reach the finals of the FA Cup and Europa League.
Solskjaer and Lingard, who split from agent Mino Raiola in November and is now represented by family members, have maintained a good relationship and the United boss has been impressed with Lingard's dedication and commitment in training despite his spell out of the team while sources have told ESPN that Lingard regularly tops the performance data stats in training.