“I don’t know how you take $54 billion out without wholesale taking out entire departments,” said Bill Hoagland, a longtime Republican budget aide in the Senate and now a senior vice president at the Bipartisan Policy Center. “You need to control it in the area of the entitlement programs, which he’s taken off the table. It is a proposal, I dare say, that will be dead on arrival even with a Republican Congress.”

Speaking to governors at the White House, Mr. Trump said his spending demands would be at the core of the speech he gives Tuesday night to a joint session of Congress. “This budget follows through on my promise to keep Americans safe,” he said, calling it a “public safety and national security” budget that will send a “message to the world in these dangerous times of American strength, security and resolve.”

In the first part of the speech, Mr. Trump will recount “promises made and promises kept,” said the aides, who requested anonymity during a briefing with reporters. The rest of the speech will focus on how he will help people with their problems and how he intends to protect the nation.

The president’s budget proposals — which were short on detail but are said to exempt not just Medicare and Social Security but also veterans’ benefits and law enforcement efforts — would lead to deep reductions in federal programs that touch millions of lives. The White House signaled that it would begin with agencies like the Environmental Protection Agency, the Internal Revenue Service and social safety-net programs.

A budget with no entitlement cuts and one that does not balance most likely has no chance of passing the House, and could be rejected by Senate Republicans as well. Mr. Trump’s proposals are too far to the right in terms of domestic cuts and too far to the left in terms of balance. Their failure could have practical implications for the White House.