The 30-year-old had left Upton Park last summer before re-joining West Ham in October on a deal that expires later this month.
He has scored three goals in 12 appearances for the Premier League strugglers and Sam Allardyce is keen to extend Cole's deal.
However, the likes of Aston Villa, where Cole spent the 2004/05 season on loan, Newcastle United and Queens Park Rangers are all reported to be monitoring his current situation.
Cole's representatives Saif Rubie confirmed discussions are ongoing with West Ham and is waiting to see what develops over the next couple of weeks.
Rubie told Sky Sports: "It is correct that West Ham have made contact about extending Carlton's deal which is due to expire later this month.
"Carlton is West Ham through and through and we will sit down and listen to what they have got to say.
"Carlton has done well for West Ham when he has played in difficult circumstances.
"Carlton wants to help West Ham get out of trouble and stay in the Premier League, but the deal has to be right him.
"There has been interest in Carlton from other clubs and he is keeping his options open.
"At this stage of his career he wants a deal best suited to him and he is West Ham player until January 20th and then we will see what happens."