Cole's future at West Ham has been the subject of speculation this month with his contract due to expire at the end of the season.
The forward is yet to be offered a new deal by the Premier League club and they could try to sell him now rather than risk losing him for nothing in the summer.
West Ham have brought in two strikers on loan this month in Marouane Chamakh and Wellington Paulista, and with Andy Carroll set to return from injury, Cole could find himself pushed well down the pecking order at Upton Park.
The news has alerted a number of teams at home and abroad to his possible availability before the close of the transfer window.
Kasimpasa have expressed an interest and they are thought to be locked in talks with the Hammers in trying to agree a fee for the former Chelsea man.
West Ham are thought to be willing to sell for a fee in the region of £3million and Kasimpasa's Turkish rivals Galatasaray and Fenerbahce are believed to be keeping a watching brief on the situation.
A number of Premier League clubs are also keeping tabs on Cole and could move for him before the transfer window slams shut next Thursday.