Chelsea winger Callum Hudson-Odoi has become the first Premier League player to test positive for coronavirus.
Arsenal manager Mikel Arteta is self-isolating after also testing positive for the virus.
Arsenal's game against Brighton on Saturday has been postponed but there has been no decision on Aston Villa's game against Chelsea on the same day.
The Premier League will hold "an emergency club meeting" on Friday to discuss future fixtures.
Chelsea say Hudson-Odoi "displayed symptoms similar to a mild cold on Monday morning" and stayed away from the the training ground.
The club added that the player is "doing well and looking forward to returning to the training ground as soon as it is possible".
Chelsea's full squad has gone into isolation.
The Gunners have closed their training ground and club staff who had recent contact with Arteta will also self-isolate.
"This is really disappointing," said Spaniard Arteta, 37. "I took the test after feeling poorly. I will be at work as soon as I'm allowed."
Arsenal expects a "significant number of people" will self-isolate, including the "full first-team squad".
The club were due to face Brighton in the Premier League at Amex Stadium on Saturday (15:00 GMT) but Brighton released a statement, shortly after confirmation of Arteta's positive test, announcing that the game had been called off.
BBC Sport understands all 20 Premier League clubs want to decide on a unified strategy, and one of the possible options that will be discussed at the meeting is postponing the rest of this weekend's scheduled fixtures.
"The health of our people and the wider public is our priority and that is where our focus is," said Arsenal managing director Vinai Venkatesham.
"We are in active dialogue with all the relevant people to manage this situation appropriately, and we look forward to getting back to training and playing as soon as medical advice allows."
Arsenal's Premier League match with Manchester City on Wednesday was postponed as a "precautionary measure" and several Gunners players went into self-isolation after Olympiakos owner Evangelos Marinakis contracted coronavirus.
Arsenal said Marinakis, 52, met a number of their players when the Gunners hosted the Greek side in a Europa League match two weeks ago.
The club said no players or staff would be tested for coronavirus.
Manchester City defender Benjamin Mendy is self-isolating as a precaution after a member of his family was admitted to hospital displaying symptoms of coronavirus.
Three Leicester City first-team players have also self-isolated after showing symptoms of coronavirus.