The announcement comes after a report on Thursday that Cole had failed to agree to personal terms with Tony Pulis's side, with some stating that the forward had been angered by Potters' chairman Peter Coates' comments that the player had injury concerns.
The Championship released a statement on Friday confirming that the player will not be moving to the Britannia Stadium, though they did not specify why the former Chelsea man's move had fallen through.
The statement read: "Carlton has not asked for a transfer and has remained committed to the club at all times.
"In being given permission to talk to Stoke, the negotiations did not break down as a result of him asking for too much money. He was discussing wage parity with other key players and was well within his right to do so.
"Further, there is no issue in terms of his reported 'injury problems'. [As an] England striker, he played 43 games last season."
The statement added: "Carlton remains part of the West Ham United first-team squad under Sam Allardyce."