As rumors circulated this week that Bill Belichick could become the new football coach at the University of North Carolina, Jamie Mottram knew that his company needed to be prepared.

Mr. Mottram is the president of BreakingT, a sports fan apparel company that specializes in viral moments. Juan Soto signs with the Mets? BreakingT went straight to its production facility for a fresh heap of “Juan Solo,” “Juan Gone” and “Do the Soto Shuffle” T-shirts. Jake Bates nails a game-winning field goal for the Detroit Lions? Perhaps BreakingT can interest you in a “Legatron” hoodie.

Keenly aware that Mr. Belichick’s unconventional move to the college ranks had the potential to set the sports world ablaze, Mr. Mottram was thrilled when Sam Franco, BreakingT’s product manager, came up with a nifty nickname: Chapel Bill, which was, of course, a play on the school’s location in Chapel Hill, N.C.

“That’s great!” Mr. Mottram recalled telling Mr. Franco. “That’s so clever.”

Yet by the time Mr. Belichick’s hiring became official on Wednesday, Chapel Bill was not merely an idea that was floating in the ether at BreakingT’s corporate headquarters in Arlington, Va. The nickname already belonged to the world — and, more specifically, to apparel shops in Chapel Hill that pounced on the news.

“It seemed like everybody had the same idea,” Mr. Mottram said.

Such was the rush to find ways to capitalize on one of the most prominent football coaches in N.F.L. history coming to a university more closely associated with basketball. The two main options: a play on his name or a reference to his addiction to short-sleeved sweatshirts.

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The department may file the suit as early as Tuesday, one of the people said.

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