The 23-year-old Croatian has been targeted by the Rossoneri, who are looking for a replacement for Hakan Calhanoglu.
But Tuttosport reports Milan could have to look elsewhere, as they are struggling to reach the demands that would satisfy CSKA Moscow, who have set a price-tag at €30m for the Croatia international.
Vlasic, who played four times and scored one goal at the Euros this summer, joined the Russian Premier League side from Everton in August 2018, initially on loan, before completing a permanent move worth €15.7m for the youngster.
He then signed a deal until June 2024 in the Russian capital and has played 104 times for the club, scoring 33 goals and contributing 20 assists in the process.
Milan are, according to the newspaper, aiming for a loan deal with an option or an obligation to buy the player, but CSKA only want a permanent deal for Vlasic.