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The N.C.A.A. basketball tournament brackets have been frozen, but, alongside whatever on-court chaos may unfold between now and early April, the coronavirus will be a persistent threat to upend the two marquee events of college sports.

If a team in the men’s or women’s tournament does not have five eligible players available for a game, its opponent will automatically advance. And although a handful of replacement squads were on standby to travel to game sites in Indiana and Texas, the N.C.A.A. said Wednesday that no new teams would be added to the tournament fields.

The men’s tournament, which will begin Thursday with four play-in games, includes 68 teams. The women’s competition, scheduled to start on Sunday, has 64 teams in its field.

“The teams have been very cooperative,” Dan Gavitt, the N.C.A.A.’s vice president of basketball, said this week in Indiana, where the men’s tournament will be held. “They understand the challenges — they’ve lived the challenges throughout the entire regular season — and so things are going quite well right now. But no one is letting their guard down. No one is making any assumptions about the lack of challenges going forward.”

Teams have been arriving for the men’s tournament since Saturday, and the N.C.A.A. said Wednesday that seven people had tested positive for the virus after about 6,900 samples were processed. Because the association requires a range of people affiliated with the tournament to test on a regular basis, it left open the possibility that few — or none — of those positive results involved players or coaches.

Although no teams dropped out before this week’s deadline for a replacement school to be invited to compete, the virus has caused turmoil around both tournaments.

Virginia, the 2019 men’s champion, will not arrive in Indiana until later this week, its plans delayed because of a positive test and contact tracing that sent most of the team into quarantine. The Cavaliers, a No. 4 seed in the West region, are scheduled to play No. 13 Ohio on Saturday night.

In the women’s tournament, Connecticut, the top seed in the River Walk region, will be without Coach Geno Auriemma for at least its first game after he tested positive for the virus. If UConn avoids an upset by No. 16 High Point, Auriemma is also expected to miss his team’s second-round game.

Although N.C.A.A. officials are imposing what they call a “controlled environment” around teams, with protocols governing everything from bus travel to testing, Mark Emmert, the association’s president, acknowledged in an interview that cases could surface during the tournaments, which are scheduled to conclude in early April.

“The first goal is no serious medical issues,” Emmert said Monday. “That doesn’t mean we won’t have teams have to pull out or somebody test positive — we’re not naïve about that — but no serious medical issues throughout.”

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