Barnes was taken off early in the second half for Cengiz Under as the Foxes failed to hold on to their early lead given them by Youri Tielemans, losing 3-1 to Mikel Arteta’s men at the King Power Stadium.
Barnes has been arguably Leicester City’s player of the season, with the 23-year-old scoring 13 goals in 35 appearances across all competitions.
But Rodgers will have to do without the former West Bromwich Albion loanee for the foreseeable future, starting with Burnley on Wednesday.
"It's not good news, it looks like he needs to go in for an operation, so it will be a minimum of six weeks," said Rodgers.
"I believe it's around his cartilage. He had a clash of knees a few weeks ago, so it has just been irritable. Today he's had a challenge right on it, has just made it worse.
"He'll need that repaired. He's had niggles with it, that's why we tried to recover him on Thursday, to take some pressure off him, but unfortunately it's a bad one for him, and a huge blow for us.
"I'll hear more on Monday, but initially that's what we think it is."
Barnes’ injury will also come as a concern for England manager Gareth Southgate, who must be considering including him in his plans for EURO 2020.
Barnes won his solitary England cap against Wales in October last year.
