Toby Alderweireld, Yerry Mina and Jerome Boateng have all been linked with a move to Old Trafford as Jose Mourinho is desperate to strengthen his defensive options, but slow progress on all three deals may force United to turn to Zouma instead.
The Frenchman, who has also been linked with a move to Everton, impressed on loan at Stoke last season but will struggle to force his way through a queue of central defenders now he is back with the Blues.
L'Equipe claims that Zouma is "in the viewfinder" of United and Everton, but Chelsea are understandably reluctant to sell to a Premier League rival.
It is believed that Everton want to take Zouma on loan, which may be a more attractive offer for Chelsea if the 23-year-old is still part of their future plans.
United would prefer a permanent deal, which Chelsea are less keen on. However, they may reconsider if a player such as Anthony Martial is included in part exchange.
Zouma played 34 Premier League matches for Stoke last season and was praised for his performances despite the Potters' relegation to the Championship.
He was a Chelsea regular during their troubled 2015/16 campaign, but he fell out of favour the following season as the Blues won the title.
The arrival of Antonio Rudiger and the emergence of Andreas Christensen forced him out of the club last summer, and with youngster Ethan Ampadu hoping to get more game time this season, Zouma's path back to the starting lineup is a difficult one.