The France international will officially become a free agent when his Monchengladbach contract expires at the end of the month and is thought to be on the radar of several top teams.
Thuram enters the free agent market on the back of registering 16 goals and seven assists in 32 games for Monchengladbach last season, including 13 strikes in the Bundesliga.
The 25-year-old also made an impression under Didier Deschamps at the 2022 World Cup, setting up two goals for France in the tournament, including Kylian Mbappe's second in their final defeat to Argentina.
Given his age, international experience and strong goalscoring form last season, Thuram will not be short of offers this summer, but according to L'Equipe, he will not end up at PSG.
The report claims that Thuram was previously intrigued by the sporting project at PSG, whose football advisor Luis Campos remained in contact with his entourage for several weeks.
However, Thuram informed PSG on Thursday that he would be seeking out opportunities elsewhere, with AC Milan currently thought to be leading the race for his services.
The Rossoneri are believed to have leapfrogged Manchester United and Liverpool in the race to sign Thuram, who is the son of legendary France defender Lilian Thuram and brother of Nice midfielder Khephren Thuram.
RB Leipzig are also believed to have emerged as contenders to sign Thuram, who prepares to leave Monchengladbach having scored 44 goals and set up 29 more in 134 matches for the club in all tournaments.
Despite conceding defeat in their efforts to sign Thuram, PSG are understood to have completed a deal for Real Madrid attacker Marco Asensio, who will join on a free transfer at the beginning of July.
Asensio will arrive at PSG to reinforce a Lionel Messi-less attack following confirmation that the seven-time Ballon d'Or winner will not be extending his contract in Paris, as he prepares to make the switch to Inter Miami on a free transfer.
The Ligue 1 champions are also thought to be actively working to offload Mbappe in the wake of receiving a bombshell letter from the Frenchman, who confirmed that he would not be triggering his one-year contract extension.
Mbappe has publicly announced his desire to stay at PSG for the 2023-24 season before making a decision on his future, but PSG are determined not to lose him on a free transfer next year.
As a result, Les Parisiens have apparently issued an ultimatum to Mbappe, whose only realistic destinations for a summer transfer are Real Madrid or Manchester United, and the latter move would only be possible if Sheikh Jassim bin Hamad Al Thani completes a takeover this summer.
PSG are also expected to confirm the signing of Manuel Ugarte and Milan Skriniar next month, while deals are in the works for Benfica's Cher Ndour and Mallorca's Lee Kang-in.