The Hammers get their campaign underway against Cardiff at Upton Park on August 17 but the former Liverpool striker is likely to sit the fixture out after failing to shake off a heel problem.
West Ham manager Sam Allardyce expects the forward to return to action in September.
“Andy is making good progress and doing some running,” he told the London Evening Standard.
“We will look at things over the next week but considering where he is in his preparation, I would have thought that we’re looking more realistically at the match following the international break, which is Southampton away on September 15.”Allardyce, meanwhile, will continue his search for another striker to replace the released Carlton Cole in the Hammers squad after he admitted he is still hoping to make another signing this summer.
Modibo Maiga is the favourite to lead West Ham's attack for the visit of Cardiff, but ex-Manchester United forward Ravel Morrison, who netted a brace in a 3-2 friendly win over Sporting Lisbon on Monday, is also in contention for a place in the squad.