The Daily Mail reports that the 19-year-old Brazilian has become frustrated by a lack of opportunities at the San Siro, and would consider a move away from the Italian giants.
Coutinho was brought to Milan by former Liverpool boss Rafael Benitez, who had high hopes for the young star from Vasco Da Gama.
The two-footed midfielder played 13 times last season, but Benitez’s subsequent departure has seen him fall out of favour with current manager Claudio Ranieri, making just a single appearance this season.
A skillful midfielder, Coutinho would add to a Liverpool midfield that has been criticised for lacking in creativity this season, despite the introduction of a number of big money signings.
Coutinho is the second Brazilian wonderkind to be linked with the Anfield club, along with 19-year-old Lucas Rodrigues of Sao Paulo.
He would also join compatriot Lucas Leiva on Merseyside, were he to make the move in the January transfer window.