The midfielder, who damaged ligaments in Thursday's Europa League quarter-final draw at Borussia Dortmund, has been ruled out for six to eight weeks.
England's first Euro 2016 game is on 11 June, nine weeks away.
Manager Roy Hodgson, who names his 23-man squad on 12 May, is reluctant to include players who are not match fit.
"I wouldn't be happy to really take players with us in the hope that they will become fit during the tournament," he has said.
Henderson, 25, started six of England's 10 Euro qualifying games, as well as the friendly win over Germany in March.
Liverpool, who have eight Premier League matches remaining this season, host Dortmund in the Europa League last-eight second leg on 14 April.
England face Russia in their first group game at Euro 2016, followed by Wales on 16 June and Slovakia on 20 June.
Liverpool captain Jordan Henderson will miss the rest of the season with a knee injury and could be a doubt for England's Euro 2016 campaign.