The Turkish sports newspaper claim that the Black Eagles are considering the offer but are yet to make a decision over the future of the Sweden U21 international.
Beşiktaş signed Milosovic for €1 million in January but he missed the second half of the league campaign with injury.
The Swedish international of Serbian origin joined Beşiktaş from AIK during the winter transfer window but broke his foot soon after making the transfer.
The powerful defender created two assists and scored three goals in 28 appearances for AIK this season before his move to Turkey.
Beşiktaş have had a busy transfer window selling last seasons top scorer Demba Ba for €14 million to Chinese giants Shanghai Shenhua.
Ba scored a career high 27 goals in all competitions for the Black Eagles last season.
The Istanbul giants have also sold 21-year-old defender Atınç Nukan to Red Bull Leipzig for €5 million.