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.
Liverpool romped to the title for the first time in 30 years last season but four straight defeats have left them in serious danger of missing out on next season's Champions League.
The Reds are sixth in the table, five points adrift of the top four.
Henderson looks set to miss Premier League games against Sheffield United on Sunday, Chelsea, Fulham and Wolves as well as the second leg of Liverpool's Champions League last 16 tie with RB Leipzig.
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