The reaction was even more intense in the Senate. Senator Mitch McConnell of Kentucky, the majority leader, had planned to use the week to publicize his floor vote to force Democrats to take a stand on progressives’ Green New Deal — an important party-building exercise for Republicans after Mr. Trump’s decision to declare an emergency at the border sparked an insurrection in their ranks this month.

Members of Mr. McConnell’s leadership team were incensed at Mr. Mulvaney and allies like the acting White House budget director, Russell Vought, for rekindling a fight that served Democrats so well in 2018 and could harm vulnerable incumbents in 2020, according to two senior aides with direct knowledge of the situation. The maneuver may make it much less likely that Mr. Vought, the chief of staff’s handpicked successor to head the Office of Management and Budget, will be confirmed by the Senate, the aides said.

And Democrats pressed their advantage.

“The equivalent of a nuclear bomb fell on our country when the president said what he said this week,” Representative Anna G. Eshoo, Democrat of California, said Wednesday as her House subcommittee began pushing through health care legislation. “This is deadly serious.”

But Mr. Trump doubled down while talking to reporters in the Oval Office. He predicted that the Texas decision would be upheld by the appeals court, then go to the Supreme Court.

“If the Supreme Court rules that Obamacare is out, we’ll have a plan that is far better than Obamacare,” he said.

[What happens if Obamacare is struck down? Read more.]

White House press aides did not immediately respond to a request for comment. And one official, who asked for anonymity to speak about the meetings, insisted that Mr. Mulvaney had simply been convening people with various views so that the president could make his own decision. But Mr. Mulvaney was described as leading the charge to back the suit, in an account of the two meetings that was described by a half-dozen people with knowledge of what took place.

Politico first reported that Mr. Mulvaney pushed Mr. Trump to get involved in the suit.

Mr. Barr did not favor the move but did not object to the White House decision once it had been made, people familiar with what took place said. And one White House official said the administration faced a deadline imposed by the court if it wanted to support the suit.