The requirement, Mr. Cuomo said, would be the subject of an executive order set to take effect on Friday that would apply to settings like buses and subway trains, sidewalks and grocery stores.

Mr. Cuomo said that face coverings, which could be masks, scarves or bandannas, could be worn below the chin when no one was nearby.

“You’re walking down the street alone?” he said. “Great. You’re now at an intersection and there are people at the intersection and you’re going to be in proximity to other people? Put the mask on.”

He added, “You don’t have the right to infect me.”

The federal Centers for Disease Control and Prevention has recommended that people wear cloth face coverings to prevent transmission of the virus, which spreads primarily through droplets generated when, for instance, an infected person coughs or sneezes.