The White House, too, said the meeting had been “productive.”

Mr. Cuomo was less bullish. “While the White House meeting was productive, it was inconclusive,” he said.

Representative Peter King, Republican of New York, said in an interview that the meeting “could not have been more positive.” He added that the only specific numbers discussed were a $30 billion price tag and a 50-50 split between the federal government and the states in picking up those costs, which has been a point of contention in the past.

The meeting came one day after Mr. Trump had sided with Mr. Schumer in agreeing to a package to provide money for Hurricane Harvey relief, fund the government through mid-December and raise the nation’s debt ceiling — bypassing his own Republican allies in Congress.

That deal was clearly on the president’s mind.

“He must have mentioned bipartisanship six or seven times, and how important he thought yesterday’s deal was,” said Mr. King, one of a handful of Republicans present.

At one point, Mr. King said the president turned to him to ask how the deal was “playing.”

“I said, ‘Very well,’” Mr. King recalled, adding that he told the president, “This project would be a perfect example of bipartisanship.”