Cole has struggled this season at Anfield, finding himself pushed to the sidelines, under the stewardships of both Roy Hodgson and Kenny Dalglish.
The England international had been linked with a move to boyhood club West Ham in the January transfer window, but it is thought that his persistent knee problems played a part in any move breaking down.
The 29-year-old signed for Liverpool in the summer on a free transfer, after Chelsea decided not to renew his contract.
Another stumbling block would be his high wages.