According to Tribuna, Galatasaray have also made enquiries about the 25-year-old, who has become a key figure in Juventus' midfield since arriving from Nice in 2024.
Khephren Thuram, the France international and younger brother of Inter Milan striker Marcus Thuram, has racked up close to 100 appearances for the Old Lady despite a string of managerial changes in Turin.
Liverpool's interest is tied to the expected departure of Curtis Jones, who is reportedly close to a move to Inter Milan.
The Reds are said to be exploring a package in the region of €45 million plus add-ons for Thuram, who would offer a different physical profile in midfield compared to Jones.
Manchester United, meanwhile, have already completed a deal for Atalanta's Ederson and are looking to add a second midfielder this summer.
Reports from Italy suggest Thuram features on a shortlist alongside Nottingham Forest's Elliot Anderson and West Ham's Mateus Fernandes, with Anderson understood to be Manchester United's preferred option.
Juventus have not set an official asking price for Thuram, but talks are expected to open in the €45-50 million range if the club decide to consider offers.
That figure represents a significant return on the roughly €20 million Juventus paid Nice for the midfielder just a year ago.
Despite the interest from England, reports in Italy have consistently suggested that Thuram himself would prefer to stay in Turin.
The midfielder has become one of Luciano Spalletti's most trusted players, and Juventus have publicly described him as central to their long-term plans.
Juventus' financial situation complicates matters. The club missed out on Champions League qualification for next season, a result that is expected to sharpen focus on player sales across the squad.
Thuram is contracted to Juventus until 2029, giving the Bianconeri leverage in any negotiation, but his current salary of around €2 million per year is modest compared to what Premier League suitors could offer.
Galatasaray's interest adds another layer to the situation, though there has been no indication of a formal approach from the Turkish club beyond initial enquiries.
For Liverpool, the calculus is straightforward. New manager Andoni Iraola needs reinforcements in central midfield, and Thuram's combination of physicality and ball-carrying ability fits the profile the club is targeting.
Manchester United face a similar equation, with the retirement of Casemiro leaving a gap in the engine room that the club have been trying to address since the start of the window.
Whether either Premier League club is willing to meet Juventus' valuation, or whether Thuram's stated preference to stay wins out, will likely become clearer as the transfer window progresses.