Cole is a free agent following his departure from the Boleyn Ground at the end of May after seven seasons with West Ham.
He made over 200 appearances for the club, scoring 47 goals, having begun his career at Chelsea over a decade ago.
The 29-year-old has also had loan spells at Charlton, Aston Villa and Wolves, and made seven appearances for England since his debut against Spain in 2009, all of which came off the substitutes' bench.
A West Ham statement on Tuesday read: "Despite the board and manager deciding to do their business early on in the window to avoid last-minute panic buys - resulting in the successful signings of Andy Carroll, Stewart Downing, Razvan Rat and Adrian - joint-chairman David Sullivan and manager Sam Allardyce did spend all night on Monday trying to sign one of three international strikers, but all opted for the lure of European football instead.
"Importantly, the board also turned down bids for several first-team players in the last week of the window as they are determined to give the squad the best chance possible of bettering last season's tenth-placed Premier League finish.
"The club are pleased to announce that popular striker Carlton Cole is also undergoing a medical today with a view to returning to the Boleyn Ground on a short-term deal."