According to the Daily Mail, Downing required 12 stitches to an Achilles heel wound following the trip to Anfield.
The newspaper also claims that the former England international has been ruled out for at least two weeks.
If that is the case, then he will definitely miss the Hammers' games against Sunderland, Tottenham Hotspur and Manchester United later this month.
Downing has made 15 first-team appearances for West Ham this season. He has yet to score a goal.