Brady started getting questions about Trump last fall when a red “Make America Great Again” cap was spotted in his Gillette Stadium locker, reportedly put there by Kraft. Brady at first gave sheepish, gee-whiz responses suggesting that it would be cool to have a golf pal as president. (“There’d be a putting green on the White House lawn,” he predicted.) But the questions took on an edgier urgency as Trump’s victories and offenses kept mounting — ending with Trump where he is now, and Brady dropping into a reflexive kneel-down every time Trump’s name is mentioned to him.

Brady, whose makeshift podium on the Minute Maid field on Monday night was swarmed with a few hundred reporters, was the central battlefield of incoming Trump-related questions. He countered with his most anodyne deflections. “I’m just a positive person” became Brady’s go-to blow off, which he also deployed in the face of several questions about Deflategate, the air-pressure-in-footballs scandal given new life this week by the prospect of Commissioner Roger Goodell’s having to award the Super Bowl trophy to the man he suspended for four games to begin this season.

Other Atlanta Falcons and Patriots players were also invited to weigh in on political matter but mostly demurred. The Falcons receiver Mohamed Sanu, a practicing Muslim, was asked to discuss Trump’s executive order banning immigration from seven majority-Muslim countries. He declined politely, other than to say that it was “ a very tough situation” and “hard for me to talk about right now.” I asked two of the more thoughtful members of the Patriots, the defensive linemen Alan Branch and Chris Long, if they had anything to say about these developments. They absolutely did, they said, and would speak out at some point later, but they did not want to create any “distractions” during Super Bowl week. The Patriots tight end Martellus Bennett did say that if the Patriots won on Sunday and were invited to the White House, he would probably not go, because “I don’t support the guy that’s in the [White] House.” And really, was that so distracting?

One enduring reason the Patriots are constantly being asked about their Trump connections is that the new president is hardly shy about reminding people of them.

He has been dropping Brady’s name — and Kraft’s, and Belichick’s — at every opportunity over the last several months. This first became evident to me in the fall of 2015 as Trump was enjoying his unlikely rise to front-runner status in the Republican primaries and I had several visits with him for a profile I was writing for the magazine. A few months earlier, I had written a story about Brady, who had just won his fourth Super Bowl and was then embroiled in the Deflategate fiasco, in which the league said it was “more probable than not” that Brady was “at least generally aware” that a minuscule amount of air pressure might have been removed from the footballs he used in the first half of the Patriots’ 45-7 blowout of the Indianapolis Colts in the A.F.C. championship game.