Stafford cleared the concussion protocol Friday, McVay said, but the Rams are still gathering information about the quarterback's neck injury.
McVay said the Rams have been taking it "a step at a time" with Stafford's availability and didn't rule out shutting down the quarterback for the season.
"I think there's so many layers to [that decision]," McVay said. "There's so much thought and dialogue that goes into it, but I think first and foremost it's got to get cleared from the medical experts. ... I'm not the expert in that arena, but if there's any sort of sense of, this is something that it'll be better served in the long term or it's going to be a few more weeks, things like that, I think that'll end up kind of taking care of itself."
McVay had said last week that Stafford felt some numbness in his legs after taking a hit in Week 11 against the New Orleans Saints, a game Stafford was playing in two days after he cleared the concussion protocol the first time.
While "Matthew's the ultimate competitor," McVay said he thinks that "in a lot of instances [the] best thing that you can do is protect him from himself."
"I know him," McVay said, noting that he and Stafford talk every day. "If there's any inkling that he can play, this guy's going to want to play. Now [there is a] possibility that we say, 'based on what we're gathering, the doctors, those types of things, the smart thing is to not play you.' And he could be upset with me, but I think he knows that I have his best interest and we have his best interest. And if that's the case, then that's definitely a possibility.
"But he's going to want to play. He's going to want to do everything in his power to be out there and compete with his teammates and do the things that's made Matthew Stafford, Matthew Stafford."