McVay said Stafford entered the protocol Tuesday after the team's medical staff did their "cleanup after the game" and "determined that they felt like that was the best thing for him."
McVay said he didn't have any indication during the game that Stafford was dealing with any concussion symptoms. He said "some of the questions" Stafford was asked after the game by the medical staff "and then what they ascertained as a result of those questions made them feel like, 'Hey, the appropriate thing for the person and the player in this instance is to put him in the protocol.'
"Whether or not that means he has a concussion or not isn't for me to truly understand," McVay said. "But there's the necessary steps that we need to take."
John Wolford is the Rams' backup quarterback and would play if Stafford has not been cleared before Los Angeles hosts the Arizona Cardinals on Sunday, but McVay said the team will "take it a day at a time with Matthew and see where we are."
"I know Matthew's going to do everything in his power to try to be able to be ready and we'll just take it a day at a time," McVay said. "But John's a guy that when he's been in these situations, he's stepped in and done a really good job. And it wasn't too long ago that he had to come in in a game that we had to have to be able to get into the playoffs and he performed incredibly well."
In Wolford's NFL debut in 2020, the quarterback helped lead the Rams to a playoff spot with a victory over the Cardinals. Playing in place of Jared Goff, Wolford threw for 231 yards and ran for 56 more in the Week 17 victory.
This season, Stafford has completed 68.4% of his passes for 1,928 yards, 8 touchdowns and 8 interceptions.
Cardinals quarterback Kyler Murray, meanwhile, appeared on Wednesday's injury report with a hamstring injury. He was listed as did not practice but the Cardinals only held a walk-through. Murray has yet to miss a game this season after missing three last season because of a sprained ankle.
ESPN's Josh Weinfuss contributed to this report.