The 25-year-old's contract with Gladbach is due to expire in June, meaning that he is now free to speak to foreign clubs about a move at the end of the 2022-23 campaign.
However, there have been suggestions that Thuram could be available for a cut-price fee in January, with his German outfit reluctant to let him depart on a free transfer.
According to the Daily Mail, Man United are 'monitoring developments' surrounding the striker, who represented France during the 2022 World Cup.
Thuram has been in strong form for Gladbach during the 2022-23 campaign, scoring 13 goals and registering four assists in 17 appearances in all competitions.
The Frenchman has managed 10 goals and three assists in 15 Bundesliga outings, while he has contributed three goals and one assist in two appearances in the DFB-Pokal.
The forward made the move to his current side from Guingamp in July 2019, and he boasts a record of 41 goals and 26 assists in 119 outings for the Bundesliga side in all competitions.
Inter Milan, Arsenal and Newcastle United are also said to be keeping a close eye on the situation surrounding Thuram, who could push for a move during the January market.
Man United terminated Cristiano Ronaldo's contract at the end of November, which has left them short of options at centre-forward ahead of a busy second half of the campaign.
Anthony Martial and Marcus Rashford are currently the team's only first-team strikers, and the former has had his injury problems during the 2022-23 campaign.
"I think we have a squad. We have the players who are matching our criteria. But with all the games that are coming, it is tough competition in all the leagues," Man United head coach Erik ten Hag told reporters last week when questioned on the team's search for a new striker.
"But if you want to also be in all the leagues and keep playing in all the leagues by winning games and cups as well. So, you need players, you need numbers to cover.
"So, we need good players. Not only numbers, you need quality players to cover that and also that you need competition. And you need also that you can make tactical choices.
"So, we are looking for the right player for our squad and has to match our sporting criteria but also our financial criteria."
Man United will be aiming to make it six wins in a row in all competitions when they welcome Bournemouth to Old Trafford in the Premier League on Tuesday night.