Kupp clutched his right ankle when he landed awkwardly trying to catch a fourth-quarter John Wolford pass and was checked by the Rams medical staff before walking to the locker room.
The Rams wide receiver, who also hurt his right ankle late in their Week 8 game but played the following week, did not speak to media afterwards.
"I don't have anything right now," McVay told reporters. "I just know it didn't look good, it didn't sound good."
Kupp, who won last year's Super Bowl MVP, has been the Rams' key offensive player this season. The wide receiver has 72 receptions for 813 yards with six touchdowns, with one rushing TD from nine carries this season.
The Rams' latest loss left the Super Bowl LVI champions with a 3-6 record, sitting last in the NFC West after three straight losses.
Kupp enjoyed an historic 2021 season, becoming the fourth player in NFL history to lead the league in receptions, receiving yards and receiving touchdowns in the same season, alongside Jerry Rice, Sterling Sharpe and Steve Smith Sr.
The 29-year-old had 145 receptions for 1,947 yards and 16 touchdowns in the 2021 regular season, before adding 33 receptions for 478 yards and six touchdowns in the postseason.