Juan Mata is the latest Red Devils player to be linked with a move away after entering the final six months of his contract at United.
It remains to be seen whether Solskjaer will bring in more reinforcements this month, with his side now serious contenders for the title after hitting top spot last week.
United have already signed winger Amad Diallo after work permit issues held up his move in October, while they continue to be linked with Dayot Upamecano as they seek a top-class centre-back.
Speculation about Jadon Sancho and Erling Haaland is never far away, although it would take serious money to prise either of them away from Borussia Dortmund.
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But as Solskjaer contemplates new signings, he also needs to ship out any surplus players as he streamlines his squad for the rest of the season.
United have already sold defender Timothy Fosu-Mensah to Bayer Leverkusen this month, with striker Odion Ighalo also likely to leave when his loan expires in less than a fortnight.
Super-agent Mino Raiola last month claimed Paul Pogba was "unhappy" and suggested he could leave in January - but the midfielder has since returned to form and is more likely to leave in the summer if he is determined to quit Old Trafford.
Solskjaer has said he could sanction "one or two" more departures - Mirror Sport looks at seven players who could be sold before the January window shuts.
Lingard has not played at all in the Premier League or Champions League this season and has started just twice in cup competitions.
He did play in United's FA Cup win over Watford earlier this month, although it remains to be seen whether he gets the nod again for Sunday's fourth-round clash against Liverpool.
French Ligue 1 side Nice were in talks with Lingard over a potential move but have since cooled their interest as the club's president, Jean-Pierre Rivere told fans not to expect "big names" this month.
The midfielder has also been linked with the likes of Sheffield United and Aston Villa this month, but it is more likely he will stay at Old Trafford until the end of the season when he will enter the last 12 months of his contract.
Rojo has been widely tipped to return to his native Argentina in the January transfer window.
The 30-year-old has not played at all this season, and while he is currently sidelined by a calf injury, he has been deemed surplus to requirements for some time.
Boca Juniors have offered the defender a two-year contract with an option of a further year, according to Argentine publication Olé.
And Estudiantes de La Plata, where Rojo started his career, are also reported to be keen to sign him after he returned there on loan last season.
Romero has been linked with a move to David Beckham's MLS franchise Inter Miami after being frozen out in the final year of his United contract.
The goalkeeper is an Argentina international but was relegated to fourth choice at Old Trafford behind David de Gea, Dean Henderson ad Lee Grant.
Romero, 33, was desperate to leave United in the last transfer window, but Everton had a £2m loan bid rejected with the Red Devils reluctant to strengthen a domestic rival.
Last month the Athletic reported that Inter Miami were ready to offer him a route out, with the MLS season due to begin in March.
Mata is another player who has entered the final six months of his contract and has been linked with a move to his former Valencia.
According to Spanish publication La Razon, the La Liga side are keen to take him back to Mestalla with Mata having increasingly become a fringe player at United.
Mata has started only six matches this season, and it looks increasingly unlikely the extra year in his contract will be triggered.
The 33-year-old could join Valencia as a free agent in the summer, although United could try and get a return on him this month.
Phil Jones is another player who looks to have no future at United, having dropped right down the pecking order following the arrival of Harry Maguire.
The former England defender has been out with a knee injury all season, but had been criticised for his performances before being sidelined.
Jones has been linked with a move to Burnley, and at 28 he could prove a bargain for another club if he can overcome his current fitness issues.
Williams broke into the first team last season and put in some impressive performances, but has found game time hard to come by this term.
The 20-year-old has been restricted to three starts in cup competitions following the return to form of Luke Shaw and the deadline day signing of Alex Telles.
Solskjaer signed left-back Telles from Porto to provide competition for Shaw, who has responded by hitting arguably the best form of his United career.
It is more likely that Williams would be loaned out for the rest of the season than sold in the January window.
James is another player who has struggled for game time with United having several forward options.
The 23-year-old winger has been linked with a move to Leeds United, with the Elland Road club having narrowly missed out on signing him before he joined United from Swansea.
As with Williams, James is more likely to be loaned out this month than sold.