Zaha sustained the injury during the early stages of last weekend's 1-1 draw with Newcastle United, although the 25-year-old was able to complete the 90 minutes.
Hodgson confirmed during Friday's press conference that the former Manchester United attacker was facing a spell on the sidelines, but the Ivory Coast international has vowed to bounce back.
After seeing a specialist over the last few days, I can sadly confirm that I will be on the sidelines for the next few weeks. Thank you to the fans for all your support and I hope to back playing again as soon as possible! 🙏🏿 #CPFC pic.twitter.com/SRQD2BA4JX— Wilfried Zaha (@wilfriedzaha) February 8, 2018
Zaha has scored 20 times in 135 appearances for Palace since returning to the London club.