Cole has been linked with a move away from Anfield after failing to establish himself under Reds boss Brendan Rodgers, making only six Premier League appearances so far this season, all of which came off the bench.
The midfielder, whose four starts this term have come in cup competitions, also saw his early campaign disrupted by a hamstring injury.
QPR manager Harry Redknapp confirmed on Tuesday he is interested in signing the 31-year-old and is planning talks with Rodgers.
But he could face competition for the midfielder's signature, with West Ham the latest to enquire and Reading also understood to be interested as they look to boost their survival hopes.
Cole began his senior career with the Hammers back in 1998 after coming through the youth ranks in East London.
The midfielder was linked with a return to his boyhood club on loan in the summer after spending last season at Ligue 1 side Lille but no deal materialised.