The 24-year-old, who joined Liverpool from Arsenal in August, was carried off on a stretcher in the 18th minute after making a sliding tackle on Roma full-back Aleksandar Kolarov.
Liverpool manager Juergen Klopp said the England international's season looked to be over.
"Yes, that is how it is, unfortunately. All the medical department is quite concerned even without a scan and the season is not that long anymore," he said.
"It’s probably a really bad injury. If we can say that already before the scan, that’s never good news so that means for us it’s a massive blow, of course," he said.
“We have still a few very important games to go and the squad doesn’t get bigger in the moment, so we obviously need to be creative in the next few games.”
England's opening World Cup game in Russia is on June 18.
(Reporting by Simon Evans; Editing by Ken Ferris)
