The Ghanaian striker, signed for just over 20 million pounds ($26.12 million) from Swansea City earlier this month, was injured during the opening Premier league game of the season, a 2-1 defeat at Chelsea on Monday.
"For Andre Ayew, there are two solutions, one with surgery and one without. We are going to decide which one soon," Bilic says.
"He could be out for two and a half months or up to four. It shouldn't be more. He could be back this year, and we just have to hope."
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On Sunday West Ham face Bournemouth in their first home league game at their new ground, London's 2012 Olympic Stadium.
They have already played a friendly against Juventus and a Europa League game there, but Bilic told reporters it will be important to get a first win.
"Hopefully that will come on Sunday," he said.
"If it's packed and we play good football, it will guarantee the atmosphere will be great."
The club are expecting a crowd of 57,000 for the game after the local council imposed restrictions, but they hope to reach 60,000 shortly.
The capacity at Upton Park, which the Hammers left after 112 years at the end of last season, was under 35,000.
"We have overseen one of the most successful stadium moves in the history of football," vice chairman Karren Brady told Sky Sports News.
"The feedback we've had is that people really like it."
($1 = 0.7656 pounds)
(Reporting by Steve Tongue, editing by Neil Robinson)
