Collins was replaced by teenager Reece Oxford in the second half of Wednesday's superb 1-0 win over Premier League title hopefuls Tottenham.
The Wales defender will now sit out of Saturday's trip to Everton, next weekend's FA Cup quarter-final at Manchester United and potentially the Premier League encounter at Chelsea on March 19.
"James got injured on Wednesday and that is a big blow because he has been marvellous," boss Bilic told a news conference on Friday.
"It's not one week or a few days, but it looks like he will miss three games."
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Bilic also confirmed that Winston Reid and James Tomkins are still unavailable, with the former in line to face United next weekend and the latter set to return for the Chelsea clash.
That means 17-year-old Oxford could step in again at Goodison Park and Bilic had an interesting way to describe the youngster's fearless approach.
"You bungee jump when you are 18. You don't at 47," he added.
West Ham, who should have Enner Valencia and Victor Moses back for the United game, are now just a point adrift of the Champions League places following their win at Tottenham, but Bilic is keen for his players to keep their feet firmly on the ground.
"It was a great win on Wednesday, but we can't enjoy it, not even for a few days, because we have a game [on Saturday]," he said.
"We've put that game to bed and we're concentrating on Everton. It will give us even more confidence of course."