The N.F.L. has postponed Sunday’s highly anticipated game between the New England Patriots and the Kansas City Chiefs to Monday or Tuesday after Cam Newton, the Patriots’ star quarterback, and the practice squad quarterback on the Chiefs tested positive for the coronavirus.

“In consultation with infectious disease experts, both clubs are working closely with the N.F.L. and the N.F.L.P.A. to evaluate multiple close contacts, perform additional testing and monitor developments,” the league said in a statement. “All decisions will be made with the health and safety of players, team and game day personnel as our primary consideration.”

The delay of one of the marquee matchups of the young season raises fresh questions about the league’s efforts to play a full slate of games ending with the Super Bowl in February, and to do so without a closed community for teams, like the N.B.A. used, to reduce the risk of infection substantially.

The league has followed Major League Baseball and relied instead of frequent testing, reconstructed team facilities to encourage social distancing, and protocols for how players, coaches and teams can interact in locker rooms, team planes and sidelines on game day.

The success of the strategy relies heavily on players, coaches and team personnel self-policing their behavior by returning home after work and not engaging in risky activities. Still, even the most cloistered players are allowed to interact with their families and others outside the N.F.L., increasing the odds of becoming infected.

The league got through the first three weeks of the season largely unscathed. But this week, the league was forced to grapple with an outbreak on the Tennessee Titans that led to their game on Sunday against the Pittsburgh Steelers to be pushed back to Oct. 25. At least a dozen members of the organization — including as many as nine players — are known to be infected.

The league’s arrival at that decision showed how difficult it is to contain the virus. After three players and five staff tested positive, the league said the Titans-Steelers game would be pushed to Monday or Tuesday. Within 24 hours, more players and staff members tested positive and the game was pushed back by several weeks. Several more members of the team have since tested positive, which has prevented the team from practicing.

In response, the league tightened guidelines and testing protocols. But given the fickle nature of the virus, it is possible the Patriots-Chiefs game could be pushed back further if more players test positive.

The Patriots confirmed a positive test, but did not identify the player, in accordance with privacy policies. In a statement released Saturday, the team said the player had been isolated and that subsequent tests done on players and staff who had been in contact with him had come back negative.

According to the league’s daily transaction wire, Newton and Jordan Ta’amu, a quarterback on the Chiefs practice squad, were added to the Covid-19 injury list on Saturday.

“We are in close consultation with the N.F.L. as well as our team of independent doctors and specialists,” the Patriots’ statement said, “and will follow their guidance regarding our scheduled trip to Kansas City and game against the Chiefs. The health and safety of our team, as well as of our opponent, are of highest priority.”

The game was to be a closely watched showdown at Kansas City’s Arrowhead Stadium featuring two of the league’s best quarterbacks. The Patriots are 2-1 with Newton, who replaced Tom Brady after he left for the Tampa Bay Buccaneers. Patrick Mahomes, the Super Bowl most valuable player, and the Chiefs are 3-0.

After the game was postponed, Tyrann Mathieu of the Chiefs wrote on Twitter, “Wear your mask, wash your hands.”

Players, coaches and staff members who test positive are not allowed in team facilities. Because of the scope of the outbreak in Nashville, Tennessee’s team facilities have been closed since Tuesday and all in-person activities halted. The Titans reportedly had new cases on Friday and Saturday, raising questions about whether the outbreak would force the postponement or cancellation of their next game, at home against the Buffalo Bills on Oct. 11.

Now, with positive cases on two more teams, the N.F.L. is facing its first serious challenge to completing the regular season according to its design. League officials have said for months that they expected players and others to test positive. Though they have said they intend to play a full 16-game season, there are several bye weeks built into the league’s calendar, which means games could be shuffled further or even eliminated as circumstances develop.

Commissioner Roger Goodell has also tried to keep players and coaches in line. Five head coaches have been fined tens of thousands of dollars for not wearing masks on the sidelines. And after the Titans’ outbreak began, the league stepped up enforcement measures to help limit exposure risks and ramp up punitive measures for missteps.

The league sent a memo on Friday to teams stating that players and coaches would be prohibited from leaving their team’s city during bye weeks. Two days earlier, Troy Vincent, N.F.L.’s executive vice president for football operations, had sent another memo that threatened suspensions or the forfeiture of draft picks to teams whose coaches did not wear masks while on the sidelines. “If we are to play a full and uninterrupted season, we all must remain committed to our efforts to mitigate the risk of transmission of the virus,” Vincent said.

Though this week marks the first widespread reporting of positive tests in the N.F.L., the league has been previously affected by isolated cases. After the season-opening game on Sept. 10 between the Houston Texans and Chiefs in Kansas City, Mo., 10 fans who had attended the game had to quarantine because they were exposed to another fan who had tested positive in a suite at Arrowhead Stadium. Kansas City has continued to allow up to 17,000 fans to attend each game — roughly 22 percent of the stadium’s capacity.

The Atlanta Falcons placed starting cornerback A.J. Terrell on the reserve/Covid-19 list before the team’s Week 3 game against the Chicago Bears, which was played as scheduled after neither team reported any additional positives.