The governor said the two men, whose relations during the virus outbreak have swung from antagonistic to conciliatory and back, discussed the role the federal government could play in increasing testing capacity, and whether more stimulus money could be made available to New York and other financially battered states.

Mr. Trump was supportive on both fronts, Mr. Cuomo said.

“The meeting went well and I think it was productive,” the governor said during a telephone interview on MSNBC. “The big issue was testing.”

At a news conference, Mr. Cuomo said the president had committed to helping New York State double its capacity for both the coronavirus and the antibody tests to 40,000 a day. He did not specify when the state would achieve that goal.

“It will take several weeks at best,” Mr. Cuomo said. “This is an enormous undertaking.”

The governor noted that it had taken the state about a month to get to where it could do 20,000 tests day, which he called the “current system at maximum.” He and other governors have said that increasing testing capacity was critical to lifting lockdown restrictions that have brought many aspects of life to a halt.