Mukiele, 24, is out of contract next year and the German club are reportedly keen to recoup at least £8.5 million for the defender.
United are looking to sign a new right-back with Aaron Wan-Bissaka having fallen out of favour at Old Trafford this season.
According to Foot Mercato, the Red Devils have been following Mukiele for years and could make a bid for him this summer.
However, they will face competition from Atletico Madrid and AC Milan who are also interested in his signature.
Mukeiele has been a mainstay for RB Leipzig since joining the club from Montpellier in 2018 for a fee believed to be over £13 million.
In 145 appearances for the Bundesliga side, he has scored 10 goals. Meanwhile, he made his debut for France last September in a World Cup qualifier against Finland.
United’s interest in signing Mukiele may be a consequence of the club’s reliance on Diogo Dalot at right-back this season.
The Portuguese has taken the place of Wan-Bissaka but has been part of a defence which has conceded 56 goals in the Premier League, the worst record in the top 12.
With the Manchester giants nearly five years without a trophy and set to finish outside the top-four this year, much of the blame has been laid at the defence.
But after their 4-0 defeat to Liverpool last month, Rangnick suggested United could need up to 10 new players across several different positions this summer.
With a limited budget, it remains to be seen which areas incoming manager Erik ten Hag decides to focus on in the market with Declan Rice, Kalvin Phillips and Darwin Nunez all reported targets for United in midfield and attack.
However, the absence of Champions League football next season will make the club a harder sell to prospective signings.
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