England international Cole was forced off during the Hammers's FA Cup Fifth Round draw with Middlesbrough at the weekend.
However, an X-ray has revealed that the former Chelsea striker has not broken an ankle, and he could return to first-team action in as little as two weeks.
The news is not as encouraging for strike-partners Boa Morte and Ashton.
The Hammers's Portuguese forward is set to be out for a month with a groin injury picked up against Middlesbrough, while Ashton is not expected to play again this season having had further surgery on his long-standing ankle problem.
NEWS: Ashton forced to undergo more surgery
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