The 25-year-old has been in impressive form for his German club during the 2022-23 campaign, scoring 10 goals and registering three assists in 15 Bundesliga appearances, while he has also netted three goals and contributed one assist in two DFB-Pokal outings.
Thuram's contract with Gladbach is due to expire at the end of June, though, and it appears unlikely that he will be signing an extension with the Bundesliga outfit.
According to Italian journalist Rudy Galetti, Inter Milan, Bayern Munich and Juventus are expressing an interest in the forward, with Gladbach prepared to sell him for the right price at the start of 2023.
However, the report claims that Man United are also in the hunt for the Frenchman, who could be available for as little as €10m (£8.7m) in January.
Thuram, currently with the France squad at the 2022 World Cup, started his professional career with Sochaux before making the move to Guingamp in 2017.
The forward spent two years with Les Guingampais before arriving at Gladbach, and he has now represented the Bundesliga side on 119 occasions, scoring 41 goals and contributing 26 assists.
Last season, the Frenchman only managed three goals in 23 appearances in all competitions, although he missed a lot of football in 2021-22 through injury.
Thuram has returned to form this season, and he could now potentially be on his way out of Die Fohlen when the winter window opens for business.
Man United had allegedly not been planning to sign a new striker during the January transfer window, with the club spending more than they had budgeted for over the summer.
However, Cristiano Ronaldo has now been released from his contract at Old Trafford, with the 37-year-old becoming a free agent.
As a result, the Red Devils are now light at centre-forward, with Anthony Martial struggling with a number of injury problems during the 2022-23 campaign.
Man United have also been credited with an interest in their former player Memphis Depay, who is expected to leave Barcelona at the start of 2023.
Meanwhile, the club continue to be linked with PSV Eindhoven's Cody Gakpo, although any move for the Dutchman is unlikely to take place until the end of the campaign.