The English worldwide will now miss Liverpool's second leg against Roma where they hold a three goal lead and would subsequently miss the final if Liverpool were to make it.
His injury leaves Liverpool with just three fit senior central midfielders - Jordan Henderson, James Milner and Georginio Wijnaldum.
"The extent of the injury means Oxlade-Chamberlain is set to miss the remainder of the campaign for Liverpool, as well as the World Cup in Russian Federation with England", Liverpool said in a statement.
Liverpool did not confirm exactly how long the 24-year-old will be sidelined but explained the damage is severe enough to rule him out of the rest of the Reds' 2017-18 campaign and the World Cup.
The 24-year-old midfielder was assessed on Wednesday and Liverpool said he will not return to full fitness until next season.
Speaking during a post-match conference at Anfield, Klopp told BT Sport: "Oxlade, it's probably a really bad injury". The player has made 42 appearances for Liverpool after signing from Arsenal on the last day of last summer's transfer window.
Oxlade-Chamberlain went down under a challenge in Liverpool's 5-2 defeat of the Italian side on Tuesday and was stretchered off after 18 minutes.
Manchester City left-back Benjamin Mendy, who returned to action at the weekend after missing the bulk of this season with cruciate ligament damage, tweeted Oxlade-Chamberlain with a message of consolation and encouragement.
Liverpool lie third in the Premier League with three games remaining - three points above Tottenham having played a game more.