The Brazilian playmaker went off injured during the Reds' 2-2 draw at Swansea on Monday, and manager Rodgers admitted it is a blow as he looks to maintain his side's spot at the top of the table.
However, after having surgery on Thursday, Rodgers expects the 21-year-old to be unavailable until the end of October.
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"He's only just having it today so the timescale from the medics is obviously up to six weeks," the Liverpool boss told reporters.
"We wouldn't anticipate any sooner than that. It's unfortunate for him because he's been in great form since he came here in January.
"Hopefully we'll get him back sooner rather than later and then he can come into the group again.
"It was one where it was more the fall than the challenge. The challenge was probably a bit clumsy but it was actually more the fall that did the damage.
"In those scenarios the bone may be bruised but unfortunately for Philippe the bone came through the muscle. That's what the pain was and why the operation was needed.
"It's unfortunate because he was a really pivotal player for us in the game. When he went, we lost that connection between our midfield and the front line; it disrupted our balance a bit."
More positive for Rodgers is the news that Daniel Agger may be able to return to the squad for Saturday's clash with Southampton after missing the trip to Swansea with a side strain.
"We'll see how he is in the next 24 hours," Rodgers added. "It's too early to say if he'll be ready for the weekend or not.
"It was just unfortunate it happened the day before the game. We're hopeful he'll be fine by the weekend."