Oxlade-Chamberlain, 26, started in both of Liverpool's game during their 2019 FIFA Club World Cup participation, including Saturday's final against Flamengo when they won 1-0 in injury time.
The England international came off after 75 minutes in the final following an ankle knock, and Klopp told reporters on Tuesday that he won't return before the end of the year, per the Mirror's David Maddock:
Oxlade-Chamberlain sat out the bulk of last season after tearing his anterior cruciate ligament in April 2018, returning to action in February 2019 and playing just 19 minutes of the Premier League campaign.
He's played a much more prominent role this season and has made 21 appearances across all competitions, netting five goals in the process.
Roberto Firmino struck in the ninth minute of extra time to bag the only goal during the final in Doha, Qatar, securing Liverpool's maiden Club World Cup:
Oxlade-Chamberlain challenged for a header in the second half but landed awkwardly on his right ankle and had to leave Khalifa International Stadium on crutches.
Klopp appeared uncertain as to how long he'll be without his creative star, per the Telegraph's Chrs Bascombe:
The Merseysiders made do without injury doubt Georginio Wijnaldum during their Club World Cup run, but the fresh concerns over Oxlade-Chamberlain will rob them of another midfield asset.
Liverpool host Sheffield United on Thursday, January 2 in their first fixture of 2020, shortly before they're at home to Everton on Sunday, January 5 in the FA Cup third round for the second Merseyside derby of the season.
Oxlade-Chamberlain was an unused substitute when the Reds thrashed their rivals 5-2 at home in the league in early December, and he'll be eager to ensure he doesn't miss another chance to face the Toffees.