Press Association Sport understands both West Ham and QPR are interested in signing the 31-year-old.
He would be available on a free transfer this month as Liverpool are prepared to cancel his contract.
The club would also pay him a negotiated settlement in order to get his estimated £100,000-a-week salary off the wage bill.
That would then give them more freedom to look to the loan market for a short-term move in the current window.
Cole joined on a free transfer from Chelsea in the summer of 2010 but was sent off on his first appearance and things never really improved much for him.
He went to French club Lille on loan last season and did relatively well but returned to Merseyside in the summer insisting he was determined to prove himself to new manager Brendan Rodgers, having already out-lasted Roy Hodgson and Kenny Dalglish in his second spell at Anfield.
However, Rodgers has not started Cole in the Barclays Premier League this season and the player has been told he has no future at Anfield.