Zaha sustained the injury during the early stages of last weekend's 1-1 draw with Newcastle United, but 25-year-old was able to complete the 90 minutes.
On Wednesday, it was reported that the former Manchester United attacker was facing four weeks on the sidelines with the issue, and now Hodgson has confirmed that is indeed the case.
"It's amazing Wilf was able to play so much of the game [against Newcastle] with the injury. He's a very quick healer and we'll be working to get him back as soon as possible. He won't play against Everton though," Hodgson told reporters.
"If he recovered in a week or two it would be a fantastic effort, but it's more realistic we'd have to wait a month for him. It's not certain. He'll be pushing the medical staff to come back as soon as possible. Quite frankly I can't remember a time when I've had this many players injured at a club."
Zaha has scored four times and registered two assists in 20 Premier League appearances for Palace this season.