Vardy supported the injury during Saturday's 1-0 win over Leeds.
The previous England striker was making his initial beginning of 2022, having been sidelined for over two months with a hamstring issue.
"He got a thump to his knee, however ideally in the following three to about a month it answers OK," said Rodgers.
"He will be out for half a month, which is a disgrace."
Vardy is the Foxes' joint top scorer with 12 objectives from 24 appearances this season.
Patson Daka, Kelechi Iheanacho, Ademola Lookman and Ayoze Perez, every one of whom are fit, are the other assaulting choices accessible to Rodgers.
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