Mr. Trump confessed that he had not personally driven in a while, and that it had been years since he had ridden the subway.

“It’s been a long time. It’s been a long time,” he said. “I know the subway system very well. I used to take it to Kew-Forest School, in Forest Hills, when I lived in Queens. And I’d take the subway to school. Seems a long time ago. I’d take it from Jamaica, 179th Street. Jamaica, right? To Forest Hills. I understand the subway very well. I used to ride between the cars.”

Referring to his parents, he added, “They weren’t thrilled when they heard that.”

Mr. Trump, at the advice of Speaker Paul D. Ryan and Vice President Mike Pence, had intended to tackle health care first — gaining hundreds of billions of dollars in cost savings — that would then be rolled into an ambitious tax cut package. Infrastructure, according to Transportation Secretary Elaine Chao and other top administration officials, was something the administration would handle when the other two bills were passed.

But the health care debacle has scuttled the president’s plans, and he is considering merging one or all of the proposals in the hopes of cobbling together a coalition. The idea of merging a big infrastructure plan with a tax proposal is not new: Shortly after taking office in 2009, President Barack Obama proposed a similar idea — and his stimulus package included more than $300 billion in tax breaks intended to spur infrastructure and capital investments to stimulate growth.

Mr. Trump is confident he will attract Democratic support — even though he has not spoken with the minority leader, Senator Chuck Schumer of New York, for weeks. He said Democrats were “desperate for infrastructure — that’s the thing that they want.”

Mr. Schumer said he was eager to start negotiations but not especially hopeful. “He hasn’t put out anything specific yet,” the senator said.