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The 47-year-old United boss wants another player to challenge Aaron Wan-Bissaka for the right-back position as well as provide an alternative attacking threat.
And The Daily Telegraph reports that Trippier has been identified as a potential target for the Red Devils.
The England international, who spent four years at Spurs, will see his current deal at the Spanish club expire in 2022.
The Telegraph claims that Trippier has 'long-term admirers' at Old Trafford and could make a move for the Atletico right-back as his contract winds down.
Wan-Bissaka, who was signed from Crystal Palace in 2019 for an initial fee of £45m, has been widely praised for his defensive strengths.
However, the 23-year-old United full-back has been criticised for not providing the attacking threat that Solskjaer needs from him.
United star Bruno Fernandes had backed Wan-Bissaka earlier this season to provide more of an attacking output for the Red Devils.
The 26-year-old Portuguese midfielder told United's official website: "I'm happy for him, he deserved to involved in these situations.
"We want him to be more positive, more in the box, because he has such great qualities.
"Everyone knows him as a defensive player but he can show more as an offensive player. I think he has the quality to do that."
United are fifth in the Premier League and are five points behind league leaders Tottenham, though the Red Devils do have a match in hand.
Solskjaer's men travel to Germany for their crucial Champions League clash against RB Leipzig on Wednesday.
The Red Devils will return to Premier League action on Saturday when they host rivals Manchester City at Old Trafford.
Featured Image Credit: PA
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