The 25-year-old is the son of former France international Lilian Thuram and the younger brother of Inter Milan forward Marcus Thuram, and came through Monaco's academy before establishing himself at Nice, where he made 167 appearances.
Thuram signed for Juventus in July 2024 on a five-year deal worth €20million, joining his brother in Serie A, and has since become a regular under successive coaches at the Turin club, also earning senior caps for France.
It was previously reported that Thuram has turned down interest from Saudi Arabia, preferring to hold out for a move to Liverpool or Manchester United instead.
Claims from Italy last week suggested Nottingham Forest have already made a concrete approach, with Juventus willing to sanction a sale for a fee in the region of €40m.
According to Tuttosport, the interest from the three Premier League clubs comes as Juventus work to finalise a separate deal to bring in free agent Franck Kessie.
Kessie, formerly of AC Milan, has been without a club since leaving Al-Ahli on a free transfer on 30 June, and is keen to return to Italian football, with Juventus having shown strong interest in re-signing him.
Luciano Spalletti's side are said to be offering Kessie a three-year contract worth €5m per season, and are hopeful no rival club enters the race, though they would first need to free up space in midfield to accommodate him.
That context is understood to be linked to Juventus identifying Thuram as a player they are prepared to sell, with the Turin club now working to finalise his departure.
United, Liverpool and Forest have all shown genuine interest in the Frenchman, and Juventus are keen to see him move to the Premier League this summer.
There has been talk of Juventus holding out for as much as €50m for their midfielder, a figure none of his suitors currently look willing to meet, with reports last month having pointed to a lower valuation of around €40m.
Any sale would help fund Juventus's move for Kessie as the club looks to reshape its midfield options ahead of the new season.