The 23-year-old is believed to have been watched by Liverpool scouts this season.
West Ham’s sudden drop in form has coincided with Lukasz Fabianski’s hip injury and since the no.1 underwent surgery on a torn hip muscle in October, the Hammers have lost six of their last eight Premier League games.
Second-choice goalkeeper Roberto attempted to fill the void left by Fabianski, however, a string of errors led to him being dropped for David Martin, who joined on a free transfer from Millwall in the summer.
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Turkish newspaper Aksam claim Trabzonspor goalkeeper Cakir, who has been capped twice by Turkey, could be brought in to give Fabianski some much-needed competition when he returns to fitness.
While Martin impressed in his West Ham debut, keeping a clean sheet in a shock 1-0 win at Chelsea last weekend, which reduced the boyhood Hammers fan to tears at full-time, he is not seen as a long-term option.
Last month, Cakir’s agent, Engin Kirkpinar, claimed Liverpool, along with three other English clubs, were tracking his client’s progress.