For a president who likes deadline-driven crises, the budget accord could bring an unhelpful peace to Washington. Beyond setting spending levels for two years, it suspends the statutory debt limit until March 2019.

The White House budget office said on Friday that in light of the agreement reached in Congress, it would modify its request to reflect the new budget cap levels, backfilling spending to catch up to Capitol Hill. It will also come with an “addendum” that will offer guidance to “account for the increased spending caps in a responsible manner.”

Mr. Trump’s budget addendum will also include a “limited set” of administrative priorities and ideas for fixing what the administration considers budget gimmicks that it says are being used to circumvent the spending caps.

The budget release will put Mr. Mulvaney, a former Republican House member from South Carolina, back in the spotlight. Mr. Mulvaney, who was a hard-line deficit hawk in Congress, has at times strained to defend Mr. Trump’s deficit-busting policies and the president’s unwillingness to cut the drivers of spending in an aging society, Medicare and Social Security. On Monday, he will have to make the case for even paying attention to the White House budget.

Analysts said it was difficult to see how the White House’s budget could be an influential or even particularly relevant document, given the vast disparity between its line-by-line allocations and the new spending caps set by Congress.

“I just don’t know how you can have a big influence when your totals add up to something so different than what Congress has set,” said Jason Furman, a former economic adviser to President Barack Obama. “This seems even more dead on arrival than it would normally be.”

Mr. Furman said he was skeptical that the White House would be able to significantly adjust spending levels in the addendum to the budget, given the complexity of the exercise.