"Most likely he probably will miss this game," McVay said Monday. "And then dependent on how he responds it will just kind of be on a week-to-week basis."
McVay said he initially thought the injury appeared as though it could have been season-ending.
After evaluation, Kupp started the second half but played only one series after the training staff determined it was in his best interest to sit out.
McVay said the team was exploring the possibility of adding another receiver.
"There might be somebody that we have a history with, but we're working through those things right now," said McVay, who declined to name the potential addition. "So I wouldn't say a workout, but there might be somebody that we would bring in."
After the game Sunday, Kupp said he was "feeling all right" but was "a little stiff."
"I thought I could go out there and play," Kupp said. "I wanted to play. It's just hard not being able to go out there and compete."
Kupp said he felt no ill will toward Stewart for the illegal tackle.
"I know he wasn't doing it with any kind of malice or anything like that," Kupp said. "He checked on me during the game, when I came out there in the second half, he asked if I was good."
Kupp also was sidelined in the second half of a Week 5 victory over the Seattle Seahawks after he was placed into concussion protocol. He cleared protocol a day before facing the Broncos.
A second-year pro from Eastern Washington, Kupp has played a significant role in the Rams' top-ranked offense this season. He has 30 receptions for 438 yards and five touchdowns.