The attacker went through surgery on a long term hernia problem on Saturday.
The club wants to get him fully fit, rather than risk the issue getting worse. But Vardy could be back in training within 10 days.
Rodgers said in his post-match press conference: "He had the operation yesterday (Saturday). Jamie was supposed to have this operation before the last round, when we played against Stoke but unfortunately the surgeon wasn't able to do the surgery then, so we had to put it back a few weeks. He had it yesterday.
"The reason for doing it was because we felt it would be a week to 10 days maximum before he was back on the pitch, and then probably another few days before he would be available. That was the two-week timeline we looked at.
"He won't lose too much fitness. It's only a minor operation. We can cope without him at this point, and then he will be fit and raring to go for the remainder of the season."