“We were very pleased with the positive attitude toward recognizing the trillions of dollars of need,” Pelosi said. “Now we have an opportunity to work together in a bipartisan way.”

House Speaker Nancy Pelosi (D-Calif.) and Senate Minority Leader Charles E. Schumer (D-N.Y.) talk to the media after meeting with President Trump on Tuesday at the White House. (Evan Vucci/AP)

Back on Capitol Hill, however, Trump’s fellow Republicans cast a much more skeptical note, questioning how any infrastructure program would be financed. Many ruled out one idea favored by Democrats: rolling back the 2017 Republican tax cuts for corporations and wealthy Americans.

“That’s a nonstarter,” Senate Majority Leader Mitch McConnell (R-Ky.) told reporters.

Unlike other previous bull sessions with congressional Democrats, Trump did not invite the media, even briefly, to observe the meeting — a decision that lawmakers credited with preserving the civil tone. But it also meant that Trump’s zeal for an infrastructure deal was only indirectly shared with the public.

“I would like to do something,” Trump said, according to a Democrat familiar with the meeting who spoke on the condition of anonymity to freely discuss the deliberations. “It may not be typically Republican.”

The White House did not respond to a question about that comment. In a statement, press secretary Sarah Sanders also called the meeting productive and spoke of the need for bipartisan cooperation but made no mention of a $2 trillion figure.

“The United States has not come even close to properly investing in infrastructure for many years, foolishly prioritizing the interests of other countries over our own,” Sanders said.

The scene of Democrats sitting in the Cabinet Room arrayed around Trump, contemplating a new spending initiative of New Deal proportions, appeared to whet a shared appetite for bipartisan governance that has been rarely seen over the past two years amid incessant partisan battles — even if the sticky details went largely untouched Tuesday.

Many of the Democratic attendees rhapsodized about what might be after they returned to the Capitol, ticking off projects such as broadband connections that could be transformational to underserved rural and inner-city locales. They did so even as their colleagues, and some of the attendees themselves, brawl on a daily basis with Trump and his administration over their resistance to oversight requests.

“It couldn’t have gone any better,” said Ways and Means Committee Chairman Richard E. Neal (D-Mass.), who is engaged in a high-profile effort to examine Trump’s tax returns, one the president has vowed to fight tooth and nail.

Neal told reporters how he generated bipartisan chuckles at the meeting by mentioning Reagan National Airport’s Gate 35X, a section particularly loathed by frequent fliers, and relayed productive side conversations on issues including prescription drugs and trade. “All of a sudden,” Neal said, “there’s an opening for some of these things.”

But the question of paying for a $2 trillion plan looms. Pelosi and Schumer, who issued a joint statement saying they were “pleased [Trump] suggested $2 trillion,” said they would meet again with the president in three weeks to discuss that contentious subject.

Schumer told reporters that Trump did not rule out tax increases — an idea many Republicans are unwilling to embrace, especially with an election next year — but offered no specifics.

“The ball is in their court,” Schumer said. “We told him that, it was repeated over and over again, that unless he is willing to come up with the pay-fors for this large package, it will never get done, and he agreed.”

Even before the meeting got underway, Trump’s advisers sought to inject skepticism into the talks. Acting White House chief of staff Mick Mulvaney, speaking at the Milken Institute Global Conference in Beverly Hills, Calif., Tuesday morning, said “it would not surprise me” if the negotiations faltered and said there was a greater likelihood of a revamped trade deal with Canada and Mexico passing Congress than an infrastructure plan.

Mulvaney said the real sticking point would be the White House’s desire to change environmental laws so that roads and other projects can be built more quickly: “The disagreement isn’t even so much on funding. The issue is that it takes so long to do projects.”

Kellyanne Conway, counselor to the president, told reporters that Trump had not endorsed the idea of paying for infrastructure with an increase in the federal gas tax, a move backed by some Democrats and business groups, including the U.S. Chamber of Commerce.

“There are many different ways to pay for it,” she said. “You could have private-public partnerships. I know the Democrats will want to raise taxes. They want to raise taxes for everything.”

But Trump, who campaigned on addressing the nation’s aging infrastructure, unveiled a plan focused on public-private partnerships last year that received a cool reception from members of both parties, who questioned the viability of relying on states, localities and the private sector to cover the bulk of its $1.5 trillion cost.

A plan released earlier by Senate Democrats would have relied far more heavily on direct federal government spending but would generate much of its revenue by rolling back the Republican tax cuts that Trump signed into law in 2017. But Democrats did not press that point inside the meeting Tuesday, multiple attendees said, preferring to give Trump leeway.

“We want to hear his ideas on funding,” Schumer said, adding that Trump’s views will be a “crucial point” to moving forward on a rare potential area of bipartisan cooperation.

That, however, has left Republicans skeptical that the negotiations will lead anywhere fruitful. As recently as 2015, Congress had to scrape together a hodgepodge of funding sources to finance a five-year, $305 billion transportation bill. That debate steered well clear of hiking the gas tax — let alone income taxes on corporations or individuals.

“Everybody’s for the easy part, but the hard part still has to be debated,” Senate Majority Whip John Thune (R-S.D.) said Tuesday, calling $2 trillion “a very, very big number” and characterizing the White House meeting as “very aspirational.”

Rep. Tom Reed (R-N.Y.), a moderate member of the tax-levying Ways and Means Committee, said he was “very hard-pressed” to see any rollback of GOP tax cuts passing muster with Republicans. He said he was suspicious that Pelosi and Schumer were instead playing political games with the issue.

“What I’m afraid is happening here is that they are not going to put any proposal out, and they are going to say to the White House: ‘See, you were the one that was supposed to come up with the tax increases. And because you won’t, we’re not going to negotiate any further,’ ” he said. “That’s classic D.C. The American people are going to see right through that.”

But Neal insisted Democrats would bring real proposals to the table: “We’ll have some ideas, he’ll have some ideas, and I think that’s how we should proceed.”