The midfielder was forced off half an hour into a 2-0 home defeat to Everton last weekend and further investigation revealed the need for surgery.
"Henderson has successfully had a corrective procedure carried out on the adductor injury," said a Liverpool statement on Friday. "He will begin a rehabilitation programme immediately.
"No specific timescale is being placed upon his return. However, the 30-year-old is ruled out initially until after the March international break."
Klopp's plans have been hit by long-term injuries to centre-backs Virgil van Dijk, Joe Gomez and Joel Matip, which have forced Henderson to deputise in defence for long stretches of the season.