FOXBORO — One of the most intriguing roster questions for the Patriots comes at the most important position.

Six-time Super Bowl champion Tom Brady is a lock, of course. Ditto rookie fourth-round pick Jarrett Stidham.

The fun part comes with trying to figure out which way Bill Belichick is headed with Brian Hoyer as Saturday’s cutdown day looms.

The preference, of course, is to keep just two quarterbacks on the regular season roster. That would allow the Patriots to perhaps keep a positional player who would actually do more than a backup quarterback who might never see the field.

Stidham still has a lot to learn and some issues to iron out. But he’s shown enough to push Hoyer out of a job by the weekend.

Maybe the decision is that easy for Belichick, who isn’t afraid to pull the trigger on anybody.

Or maybe it’s not. Because the reality is that the Hoyer situation is complex and has a few layers.

For starters, Brady would love to have him stick around. He’s comfortable with Hoyer riding shotgun, enjoys having him in the quarterback room, and believes he provides valuable assistance week to week, even though he’s not actively playing in the games.

It’s also no secret Hoyer has done great work with the scout team. He was instrumental in helping the Patriots defense take down Jared Goff and the Rams in Super Bowl LIII. Hoyer had some institutional knowledge of the Rams offensive philosophies. He had a connection to Sean McVay, via Kyle Shanahan. And he played in San Francisco for Shanahan, while McVay was once a tight ends coach for Shanahan.

Patriots offensive coordinator Josh McDaniels sung his praises Monday.

“I won’t make any of those decisions (on how many quarterbacks we keep), but I love Brian Hoyer. Brian’s been great for us. He’s great for Tom,” McDaniels said. “He’s done a nice job in the preseason, obviously. Brian knows our system, knows how to operate and function within the things we do. (He) throws the ball well and is an extremely hard worker. Really smart, great teammate, and as good a leader as I’ve seen as a backup quarterback that I’ve been around. Because, you know, that’s a difficult position to be in sometimes. But Brian understands who he is and what his role is and does a tremendous job of pushing his teammates and goes in there and takes his job very seriously, whether it be in practice or in games, scout team, our offense, whatever it might be. He makes the guys around him better.”

Stidham is not at a point where he can be a factor with the scout team and help the defense crack an opponent. He’s worried about developing as a quarterback in this league.

So Hoyer still has some value in-house. He would also be better equipped to take over in a pinch should something minor happen to Brady. He could get the offense through a regular-season emergency scenario. Down the stretch, when playoff seeding is so important, he’s more likely to hold the fort and keep the wins coming. Stidham isn’t quite ready to pinch-hit and assure those wins. However, he has shown enough that with a little more time, he could handle a short-term crisis.

“I think he’s a good kid. He loves football. He’s a hard worker. He really comes in every day, he makes mistakes every day, like they all do as rookies, and he gets better,” said McDaniels. “He takes coaching. He listens well. And he can improve from one day to the next if he’ll just take the things we’re talking about in the meeting room and apply them on the field the next day.

“He’s demonstrated toughness. He’s taken some hits. He’s been accurate when we’ve given him time to throw,” McDaniels went on. “And generally speaking, he’s run the offense somewhat the way we would like him to run it. Again, there’s huge room for improvement. I don’t think there’s a rookie in the National Football League after three preseason games that you wouldn’t say that about. Our goal is to just try to get a little better every day with him.”

If Brady suffers a major injury, the backup becomes a moot point. Neither Hoyer nor Stidham is going to produce the types of must-have game-winning drives Brady did against the Chiefs in the AFC Championship Game.

So do you keep three, or two?

If they keep two, the Patriots could simply release Hoyer or trade him. The latter might be the most appealing to Belichick because he’ll get something in return. Depending how desperate a team might be, he might get a fifth- or sixth-round pick in return for Hoyer. After Andrew Luck’s stunning retirement announcement Saturday, the Colts could sure use a veteran backup. That’s one possible stop for Hoyer. There are others, and there will be more depending on how long Belichick wants to wait.

Plus, if they think Stidham might eventually be the successor to Brady, Hoyer is just in the way. Stidham needs as many reps as possible, even if it’s just in practice.

As it is, Belichick has only kept three regular season quarterbacks only twice during the past decade.

So the answer might already be known. But then again, does Belichick ever do what anyone thinks?