The 29-year-old was in strong form for his French club during the 2021-22 campaign, featuring on 37 occasions in Ligue 1, contributing five goals and 11 assists in the process.
Clauss's future has recently been the subject of much speculation, with Marseille and Chelsea believed to be keen to sign the wing-back during this summer's transfer window.
A recent report from 90min claimed that Marseille had moved to the head of the queue for Clauss, who is likely to be available for around €10m (£8.5m) in the current market.
However, according to Portuguese transfer expert Pedro Almeida, Man United are now closing in on a deal for the France international.
Clauss only made his international debut for France in March, and he has since represented the Blues on four occasions, with the defender bidding to be in the squad for the 2022 World Cup.
With just 12 months left to run on his contract, Lens are not in a strong position to demand a high transfer fee for the former Strasbourg youngster.
Clauss largely played as a wing-back for his French club during the 2021-22 campaign, and it is believed that his versatility makes him an attractive proposition for Man United.
The 20-time English champions have already improved their full-back options this summer with the acquisition of Netherlands international Tyrell Malacia from Feyenoord.
Diogo Dalot was Man United's first-choice right-back in the second half of last season, and the Portugal international is expected to stay at Old Trafford this summer.
However, Aaron Wan-Bissaka is believed to be surplus to requirements at the club, while Brandon Williams and Alex Telles have also been strongly linked with exits.
Ethan Laird is currently battling to be a first-team option for Erik ten Hag next term, with the youngster on the club's pre-season tour of Thailand and Australia.
However, it is believed that Laird could head out on loan again next term, while departures for Wan-Bissaka and Williams would leave Man United short of options at right-back.
The 20-time English champions have also improved their centre-back options with the arrival of Lisandro Martinez, who is expected to battle Harry Maguire, Raphael Varane and Victor Lindelof for a starting role during the 2022-23 campaign.
Man United have made three signings during this summer's transfer window, bringing in Malacia, Christian Eriksen and Martinez, while they continue to work on a deal for Barcelona's Frenkie de Jong.