After quiet starts to the summer transfer window, each of the Premier League giants are now making progress when it comes to strengthening their respective squads.
Chelsea have added Manchester City winger Raheem Sterling to their ranks and it is expected that at least two new centre-backs will arrive in the coming weeks.
Meanwhile, United announced on Friday afternoon that they completed the acquisition of Denmark international Christian Eriksen on a free transfer, the playmaker joining Tyrell Malacia in making the switch to Old Trafford.
However, both the Blues and United are yet to address their potential issues on the right-hand side of defence with players in that position potentially departing elsewhere.
Chelsea legend Cesar Azpilicueta may sign for Barcelona should the two clubs agree on a free, while Aaron Wan-Bissaka is being tipped to leave United before the start of September.
According to the Evening Standard, the teams may emerge as rivals for the signature of Clauss ahead of the start of 2022-23.
Since 2020, Clauss has gone from a relative unknown who was available on a free transfer to arguably the best wing-back in Ligue 1.
In two seasons with Lens, Clauss has contributed eight goals and 18 assists from 74 appearances, earning a spot in the Ligue 1 Team Of The Year for 2020-21.
Although Lens would prefer to retain the services of the 29-year-old in the long term, there are now less than 12 months remaining on his contract.
With Lens having narrowly missed out on European football for the second campaign in succession, the France international is now expected to move to a bigger club on the continent.
The report claims that United have already approached Lens regarding a potential deal for Clauss, who can also play further forward down the flank.
Lens will allegedly be prepared to accept a fee in the region of £8m for a player who is in line to feature in France's squad for the 2022 World Cup.
Atletico Madrid have also been credited as admirers of the former Arminia Bielefeld player during the current transfer window.