Wilshere suffered the injury early in England's 1-0 friendly win over Denmark on Wednesday, but played on after treatment before being withdrawn midway through the second half at Wembley.
Arsenal's league return trip to Hull City on April 19 looks the most likely date for Wilshere's comeback, less than a month before England boss Hodgson names his provisional World Cup squad on May 13.
Hodgson told Sky Sports News: "It's always sad when you have a player come from a club, use him on international duty and he gets an injury.
"It's very sad and disappointing for Jack and Arsenal but unfortunately these things happen.
"It was an honest challenge but he took a nasty kick and I can only hope he recovers as early as possible."
Hodgson was speaking at the annual London Football Legends Awards in aid of the Willow Foundation.
He was presented with a Lifetime Achievement Award with West Ham Director of Youth Development Tony Carr given the Willow Award, which recognises outstanding contributions off the pitch.
For more information about the Willow Foundation visit www.willowfoundation.org.uk