But the initiative has run into immediate opposition from Republicans who balk at the hefty price tag and from conservative allies who are pushing lawmakers to block it. Those opposed to the deal include Trump’s top aide, Mick Mulvaney, and Senate Majority Leader Mitch McConnell (R-Ky.), who is not in favor of the spending, according to people who have spoken to him.

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Mulvaney said in an interview Friday that he did not disagree with the president on the need for an infrastructure initiative and was seeking to identify at least $1 trillion for the purpose.

“Is it difficult to pass any infrastructure bill in this environment, let alone a $2 trillion one, in this environment? Absolutely,” Mulvaney said.

On Saturday morning, the president tweeted: “There is nothing easy about a USA Infrastructure Plan, especially when our great Country has spent an astounding 7 trillion dollars in the Middle East over the last 19 years, but I am looking hard at a bipartisan plan of 1 to 2 trillion dollars. Badly needed!”

Senior Republican lawmakers are also actively downplaying any prospects for an infrastructure deal of that scale this year — though Trump himself proposed the $2 trillion figure to House Speaker Nancy Pelosi (D-Calif.) and has seemed eager to pursue that objective with Congress.

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“I think we need to figure out what we need, and then we can figure out how much it costs to build it,” said Sen. John Cornyn (R-Tex.). “It seems backward to me to talk about how much we want to spend without talking about what we need and what the priorities should be.”

Senate Majority Whip John Thune (R-S.D.), the party’s main vote-counter in the Senate, called the $2 trillion number “incredibly ambitious” and said he was “still chewing on” the size of the package that Trump is seeking.

“The question most of our members are asking is, ‘How are we going to pay for this?’ ” Thune said. “We’ll see.”

The resistance from Trump’s own party illustrates how quickly the bipartisan idea has been imperiled in Washington, as the president heads into his reelection campaign with no significant legislative agenda and besieged by congressional investigations.

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The infrastructure effort also underscores Trump’s penchant for talking up deals with Democrats behind closed doors — leaving his own advisers and congressional Republicans to reel him back.

Earlier in the administration, Trump praised Sen. Elizabeth Warren (D-Mass.) — a potential 2020 foe — for her ideas because, in his view, she was determined to spend more than Republicans. He would tell aides to get a list of projects and “let’s just spend it,” in the words of one former administration official.

Trump also grew frustrated when the idea of “Infrastructure Week” became a running joke symbolizing the disorganization of his White House, current and former aides said. Trump actually wanted to embark on such an effort, seeing it as a populist idea key to his reelection chances, they said, and thought the government could make money from toll roads.

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The original White House plan was to leverage $200 billion in federal funding with larger private expenditures, but Trump always wanted to spend more. Gary Cohn, when he was the president’s national economic adviser, would argue for such public-

private partnerships, but the president’s New York real estate friends — including investor Steven Roth, who was an economic adviser to the Trump campaign — told Trump it was a harebrained idea. Trump disparaged the concept in the meeting with Democrats this week.

“Two challenges: It’s an extremely large number, and there is a complete lack of definition in what value would be generated if we made that investment,” said D.J. Gribbin, who was the Trump administration’s infrastructure policy expert until last year. “I find it unusual how much the conversation is, ‘How much we are going to spend?’ instead of, ‘What are we going to accomplish?’ ”

Gribbin said Trump liked larger numbers: “That is his management style. ‘Why can’t it be bigger? Let’s make it bigger. Why can’t we do more?’ ”

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But while Trump has tasked aides with finding $2 trillion, so far there are few concrete answers, according to White House officials. Aides said they are not sure how they would even spend that amount of money.

There’s been little follow-up from the White House to Pelosi or Senate Minority Leader Charles E. Schumer (D-N.Y.), according to Democratic aides, except for a phone call between Trump and Pelosi later Tuesday afternoon that focused on infrastructure as well as an effort to reduce the cost of prescription drugs.

Meanwhile, the powerful network of conservative organizations led by the billionaire industrialist Charles Koch has made calls against the bill, according to Capitol Hill officials.

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Finding $2 trillion in federal funds is indeed a tall order.

According to a December analysis from the Congressional Budget Office, raising fuel tax rates by 35 cents and pegging them to rise with inflation would generate only about a quarter of the necessary revenue over 10 years. Since 1993, the rates have remained at 18.4 cents per gallon of gasoline and 24.4 cents per gallon of diesel fuel, and recent Democratic proposals to raise them have been far less ambitious.

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Getting the remaining $1.5 trillion would involve much more significant tax increases, and Democrats have not been shy about eyeing the recent Republican tax cuts that disproportionately benefited corporations and wealthy individuals. But even fully reversing the corporate income tax cut, which dropped the rate from 35 percent to 21 percent, would not close the gap, and many Democrats are unwilling to go that far.

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Though Democrats said Trump did not rule out tax increases in their meeting this week, multiple GOP leaders have made clear they do not support rolling back their marquee 2017 tax law in any way.

“It took us three decades to do it,” House Minority Leader Kevin McCarthy (R-Calif.) said Thursday during a Washington Post Live interview. “I don’t want to touch it.”

Several Republicans, including McCarthy, have pointed to a recent bipartisan bill called the Generating American Income and Infrastructure Now Act, which would sell off troubled Agriculture Department loans to fund infrastructure and reduce the deficit. But that bill is expected to raise only tens of billions of dollars, nowhere near the four-comma figure Trump and Democrats are eyeing.

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Many GOP figures are simply calling the task impossible.

“I haven’t seen any set of pay-fors that would come anywhere close to $2 trillion that Republicans and Democrats in Congress can agree upon,” House Minority Whip Steve Scalise (R-La.) said.

Still, Grover Norquist of Americans for Tax Reform said that whatever heartburn the $2 trillion figure generated among conservatives, it was accompanied by an even bigger wave of relief that Trump had not committed to any particular tax increase.