The 25-year-old striker will become a free agent when his contract at Bundesliga club Borussia Monchengladbach expires at the end of this month.
A host of top European clubs are said to have been alerted to Thuram's situation, with Man United understood to have made contract with the player's entourage earlier this year.
Erik ten Hag is keen to delve into the summer transfer market and recruit a new centre-forward, with Thuram thought to be a cheaper alternative to the likes of Harry Kane, Victor Osimhen and Randal Kolo Muani.
Liverpool are also thought to be considering a summer swoop for the soon-to-be free agent, who scored 13 goals and provided six assists in 30 Bundesliga matches for Monchengladbach last season.
Jurgen Klopp is prioritising midfield additions at present, but the Reds may decide to bolster their attacking options following the departure of Roberto Firmino.
The likes of Juventus, Inter Milan, Bayern Munich, Newcastle United, Chelsea and Barcelona have also been credited with an interest in the striker, although the latter two are believed to have pulled out of the running as they prioritise other targets.
Nevertheless, according to Sky Sports Germany reporter Florian Plettenberg, AC Milan are currently in pole position to recruit Thuram despite late interest shown from RB Leipzig.
Plettenberg claims that Milan have offered Thuram a signing-on fee in excess of €10m (£8.6m) as well as a salary of around €5m (£4.3m) per season, not including potential bonuses.
The Rossoneri are on the lookout for a new striker after 41-year-old legend Zlatan Ibrahimovic announced his retirement from football last month.
While Rafael Leao was Milan's standout attacker operating off the left last season, 36-year-old Olivier Giroud ended the 2022-23 campaign as the club's top scorer with 18 goals in 47 games across all competitions, two more than Leao.
Meanwhile, Ante Rebic struggled to make his mark as he endured an injury-hit season, and both Divock Origi and Charles De Ketelaere failed to reach the heights expected after moving to San Siro last summer, scoring just two goals in 19 Serie A starts between them.
As for Leipzig, they are thought to have identified Thuram as a potential replacement for fellow Frenchman Christopher Nkunku, who officially completed his big-money move to Chelsea earlier this week.
Thuram spent four years at Monchengladbach, scoring 44 goals and registering 29 assists in 134 appearances across all tournaments, while he featured five times for France at the 2022 World Cup.