Alexander-Arnold picked up the knock during the game and has been undergoing tests to ascertain its seriousness.
Liverpool have a break after the game tomorrow until January 21, when Premier League action will resume against Bournemouth at the Vitality Stadium.
Another injury is the last thing the Reds want right now, with Mohamed Salah and Wataru Endo on international duty at the African Cup of Nations and Asian Cup.
Andrew Robertson, Kostas Tsimikas, Dominik Szoboszlai, Joel Matip, Thiago Alcantara and Stefan Bajcetic are also sidelined, and manager Jurgen Klopp will hope Alexander-Arnold has not sustained a serious injury.
Conor Bradley is expected to be deployed at right-back against Fulham tomorrow after he impressed against Arsenal when he came on as a substitute.
The Northern Ireland international will be looking forward to more game time with Alexander-Arnold likely to miss more than a game.
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