Vardy, 34, suffered the injury in a 1-0 win over Liverpool on December 28 and scans have revealed the problem is worse than first feared.
"Jamie will be out for a number of weeks. It's been unfortunate for him but we're looking at up to eight weeks," Rodgers said on Friday.
"His is the higher end of his hamstring -- you are looking into March for Jamie. He doesn't need an operation and that was the only good news from it."
Leicester have had three Premier League matches postponed in recent weeks, two of which were called off because they were unable to field a team due to injuries and a coronavirus outbreak.
The FA Cup holders begin their defence of the trophy at home to Watford on Saturday but Rodgers' side is severely depleted by injuries and the absence of players at the Africa Cup of Nations.
Rodgers could have as few as eight senior outfield players to call upon this weekend with Jonny Evans, Patson Daka, Caglar Soyuncu, Luke Thomas, Boubakary Soumare and James Justin joining Vardy on the sidelines.
Evans has been ruled out until April due to his own hamstring injury, which required surgery.