These reports - carried by Portuguese newspaper A Bola - claim that El Shaarawy will be allowed to leave the San Siro if they club receives a fee in the region of the asking price.
Liverpool were linked with the Italy international earlier this week, with Brendan Rodgers thought to be keen to add to his attacking options in January.
El Shaarawy would be a useful addition for the Anfield club - he can play across the forward line - important considering Liverpool often play a system that emphasises attacking mobility and fluidity.
The Italian forward has featured just twice for AC Milan in Serie A this season - and is yet to open his league account.
This barren stretch in front of goal dates back to last season - El Shaarawy enjoyed a blistering start to the campaign last term with 14 goals through the New Year.
But his form deserted him in the second half of the season, and the Italian managed just two more league strikes to take his tally for the season to 16.
Still, the 20-year-old remains one of the most gifted young forwards in European football and Liverpool are not the only club credited with an interest.
The report claims Borussia Dortmund are also tracking developments, and could be tempted to rival Liverpool with a move for the attacker in the January window.