The Sun says the long-serving striker has been offered £30,000 a week plus £5,000 appearance money to stay at the club.
But Cole, 29, is a free agent in the summer and is believed to want to study his options.
A Hammers source said: "Carlton has been made an off-the-record deal but we want an answer by the end of the season.
"The deal is there until the last day of the season, after which both sides are free to do whatever they will. We believe it is a good offer."