The 35-year-old is approaching the final months of his current agreement at Anfield, which runs until June 2027. Milan believe they could have an opportunity to sign the Netherlands captain on a free transfer if he does not agree an extension with Liverpool.
The Italian club have been linked with Van Dijk previously, including during last summer's transfer window, but are now reportedly more optimistic about their chances of securing his signature.
AC Milan owner Gerry Cardinale is particularly keen on the prospect of bringing the experienced centre-back to San Siro. Club advisor Zlatan Ibrahimovic is also said to have a good relationship with Van Dijk, strengthening the appeal of a potential move.
The Rossoneri are understood to recognise, however, that competition for the defender's signature could be considerable. Clubs with greater financial resources are also monitoring his contractual situation, with interest reportedly coming from the Saudi Pro League and Turkey's Süper Lig.
Milan would therefore hope to convince Van Dijk through the sporting and historical appeal of the club, rather than simply competing financially with rival offers.
Van Dijk has been a central figure at Liverpool since joining from Southampton in January 2018. He has made 389 competitive appearances for the club, scoring 36 goals and providing 16 assists.
During his time on Merseyside, he has helped Liverpool win the Premier League, Champions League, FA Cup and League Cup, becoming one of the most influential defenders in the club's modern history.
Earlier in his career, the Dutchman played for Groningen, Celtic and Southampton before making the move to Anfield.
His immediate future remains uncertain, and Milan's reported interest is dependent on whether Liverpool can agree a new contract with their captain.
Van Dijk is currently absent from the Netherlands squad for the November international fixtures after withdrawing through injury.