But the drastic reordering of government that Mr. Trump has embraced includes many measures long sought by conservatives on Capitol Hill, including adding work requirements for food-stamp eligibility and opening the Arctic National Wildlife Refuge to oil drilling. It would also eliminate whole programs, including AmeriCorps, the Corporation for Public Broadcasting, the National Endowment for the Arts and the Low Income Home Energy Assistance Program.
The budget would increase military spending by 10 percent and calls for spending $2.6 billion on border security, including $1.6 billion to begin funding a wall on the border with Mexico.
Some of the president’s proposals are likely to survive.
For Republicans, the stakes of the coming budget season go beyond the intricacies of budgetary minutiae: Republicans want to use their budget to pave the way for an overhaul of the tax code that could skirt a Senate filibuster. If they cannot agree on a budget, Mr. Trump’s promised “biggest tax cut” in history will be doomed. A protracted fight over the budget would also further delay the orderly appropriations process that Republicans have promised to follow after years of neglect.
If congressional Republicans fail to pass spending bills this summer, they again run the risk of funding the government through stopgap resolutions that keep programs on autopilot — and in the shape that President Barack Obama left them in.
“It’ll be very difficult in both bodies to pass a budget proposal,” Mr. Rogers said.
The next step for Republicans in Congress is to agree on a budget blueprint, which sets spending levels and provides a road map for spending and revenue in the coming years. But first, they must find a way to overcome their diverse views on fiscal policy.
Mr. Trump’s budget, drafted by a budget director, Mick Mulvaney, who came from the most conservative corners of the House, starts the conversation on friendly House Republican turf.
Representative Mark Meadows, Republican of North Carolina and the chairman of the conservative House Freedom Caucus, said that was the right starting point. The budget negotiation “goes from conservative to moderate, and that’s the way that it should go,” Mr. Meadows said. “If you start in the middle, you make everybody mad when you move one way or another.”
Senator Chris Van Hollen of Maryland, who was the lead Democrat on the House Budget Committee for years, was not so sanguine.
“There’s always been a divide between the House and Senate Republicans on a lot of these issues, but this looks like it was written by House Republicans on steroids, and I think it will be difficult for them to get it through the Senate,” he said.
Republican lawmakers already face a time crunch, given that Mr. Trump offered his budget three months past the statutory deadline in February.
While new presidents routinely take more time to submit their inaugural budgets, Mr. Trump unveiled his unusually late, and in an uncommonly low-key fashion, dispatching his budget director to unveil the plan while he was overseas. That raised questions about whether he would take a leadership role in the coming spending debates.
The House and Senate budget committees both expect to introduce their proposals in June, according to congressional aides. The House plan is expected to incorporate the significant changes that Speaker Paul D. Ryan, a former budget committee chairman, has long championed for Medicare, a major break with Mr. Trump, who has promised to leave Medicare alone.
For years, Mr. Ryan has tried to shift Medicare away from its open-ended commitment to pay for medical services and toward a fixed government contribution for each beneficiary — a change he has said would inject market forces and competition into the program.
Mr. Ryan told reporters on Tuesday that Congress would take the president’s budget “and then work on our own budget, which is the case every single year.”
The Senate majority leader, Mitch McConnell of Kentucky, was equally noncommittal.
“Every president since I’ve been here, and that covers a good period of time, has made a recommendation, and then we decide what we’re going to do with those recommendations,” Mr. McConnell said.
Mr. Mulvaney conceded that the plan would not be embraced in its entirety, but said it was a signal from the president to Congress about his priorities and goals.
“If Congress has a different way to get to that endpoint, God bless them — that’s great,” Mr. Mulvaney said Monday as he previewed the plan. “Do I expect them to adopt this 100 percent, wholeheartedly, without any change? Absolutely not. Do I expect them to work with the administration on trying to figure out places where we’re on the same page? Absolutely.”
Democrats came out strongly against the budget, saying it would hurt the poor and the working class. They are hoping that Republicans will brush off the White House’s requests, much as they did when Mr. Trump sought funding for his border wall as well as billions of dollars in cuts to domestic programs as lawmakers hammered out an agreement to fund the government through September.
The Senate minority leader, Chuck Schumer of New York, said Republicans “dislike this budget almost as much as we do.”
“And so the likelihood is what happened with the 2017 budget will happen here,” Mr. Schumer said. “Democrats and Republicans will tell President Trump and his minions to stay at the other end of Pennsylvania Avenue. Let us work out a budget together that will make America a better place.”