Earlier this week, it was claimed that Los Blancos were prepared to offer €20m (£17.4m) for the German's services in 2024 following their injury concerns in the early weeks of the new term.
Real boss Carlo Ancelotti is working without Eder Militao for the long term due to his ACL issue, while David Alaba has also struggled to stay fit throughout the 2023-24 campaign so far.
In addition, club captain Nacho Fernandez is out of contract at the end of the current season, and his departure would open up a defensive void needing to be filled in the summer transfer window.
Thiaw has come on leaps and bounds since leaving German crisis club Schalke 04 last summer, and the 22-year-old's contract with the Rossoneri does not expire for another four seasons.
West Ham United were also believed to be weighing up an approach for Thiaw, having seen a move for Manchester United and England centre-back Harry Maguire collapse over the summer.
However, Calciomercato states that Thiaw is in no rush to leave San Siro, and Milan are even keen to hand him a new-and-improved contract to reflect his importance to the squad.
Thiaw is said to earn a yearly €800,000 (£694,888) under the terms of his current contract, but Milan are prepared to treble his wages to as much as €2.7m (£2.4m) per annum.
The report adds that all parties will sit down for a 'calm' discussion over his contract in the spring, and any extension would see Thiaw commit his future to the club until at least 2028.
The Germany international is settled in Milan, but while he has no plans to depart the club for at least another two years, his long-term plan beyond
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