But the bureau has long been vilified by Republicans as an overreaching, aggressive arm of government.
Now, with President Trump’s support, they have intensified their efforts to curb its power. In a significant blow to the agency, Congress recently overturned a rule that would have allowed millions of Americans to band together in class-action lawsuits against financial institutions.
Installing Mr. Mulvaney, a fiscal hawk who previously called the agency a “sick, sad” joke, as the agency’s interim leader would immediately alter the consumer bureau’s focus.
Two White House officials, speaking to reporters in a briefing call on Saturday, cited guidance from the Office of Legal Counsel, part of the Justice Department, as legal grounds for Mr. Trump’s move.
Mr. Trump’s decision to name his own temporary director was a “typical, routine move,” said one official, who spoke on the condition of anonymity.
The official later said, “We have gone out of our way to avoid any unnecessary legal battle with Mr. Cordray,” adding that “his actions clearly indicate he is trying to provoke one.”
The officials said the Federal Vacancies Reform Act gave Mr. Trump the authority to override the successor named by Mr. Cordray, who resigned about a week earlier than expected. If the White House’s plan proceeds, Mr. Mulvaney would run the agency until a permanent successor is confirmed by Congress.
In an eight-page opinion released late Saturday, the Office of Legal Counsel said the president had the ability to choose a successor.
The 2010 Dodd-Frank Act, which established the bureau — and gave it unusual independence from Congress and the president — laid out a succession path in the case of the director’s “absence or unavailability,” the office acknowledged. The deputy director then steps in as the agency’s acting head, according to the act.
But that law “does not displace the president’s authority under the Vacancies Reform Act,” wrote Steven A. Engel, the assistant attorney general in charge of the office.
That is in line with two past opinions from the counsel office, addressing vacancies at the Justice Department and at the Office of Management and Budget. Those written opinions endorsed the president’s authority to use the vacancies act to override an agency’s designated succession path and appoint his own acting leaders instead.
But the consumer bureau holds a different legal view.
Mr. Cordray cited the Dodd-Frank Act when he followed up his resignation announcement on Friday with a letter naming the agency’s chief of staff, Leandra English, as deputy director. Mr. Cordray said he expected Ms. English to take over from him.
Democrats, who fought for the bureau’s creation and championed its work as a valuable defense against predatory companies and abusive financial practices, are likely to push for a legal challenge to Mr. Trump’s move.
Senator Elizabeth Warren, Democrat of Massachusetts, who proposed the consumer bureau and helped set it up, said Mr. Trump’s move was legally impermissible.
“The Dodd-Frank Act is clear: If there is a C.F.P.B. director vacancy, the deputy director becomes acting director,” she wrote on Facebook. “President Trump can’t override that.”
Former Representative Barney Frank, a co-author of the act, said the consumer bureau is “different in every way” from other federal agencies and was intended to have more autonomy.
“We gave a lot of attention to how to structure the C.F.P.B. and how to protect its independence, because its job is to go after some very powerful forces in the economy,” Mr. Frank, who wrote the law with Christopher Dodd, then a senator, said in an interview on CNN.
The agency’s director, who serves a five-year term, cannot be fired by the president except for cause — a provision that is currently under review by United States Court of Appeals for the District of Columbia Circuit in Washington. The court’s ruling is expected soon.
Mr. Trump’s decision to appoint Mr. Mulvaney is “unprecedented,” Mr. Frank told CNN. Asked how the move could be challenged, he answered, “I literally don’t know.”
Some legal experts say there is reason to question whether the Federal Vacancies Reform Act supersedes an individual agency’s designated line of succession.
The Office of Legal Counsel’s views are generally considered binding on the executive branch, but the judiciary is not constrained by them. Courts have, at times, disagreed with the office’s legal interpretations — and their rulings take precedence.
“Any Office of Legal Counsel, in any administration, would have given the same answer with respect to this issue,” said Aditya Bamzai, a law professor at the University of Virginia. “But if we see a legal challenge, the executive branch’s positions don’t always prevail in court.”
Stephen I. Vladeck, a law professor at the University of Texas, said it was an “open question” which act — the vacancies law or Dodd-Frank — should take precedence, especially because Dodd-Frank was enacted more recently.
On a mailing list for administrative law professors, a debate broke out on Saturday over specifics of the two laws. Experts argued about the nuances of various bits of fine print, including how individual words like “shall” and “may” should be interpreted.
On Twitter, one of the discussion’s participants, Marty Lederman, said that this “might well be the case to test” the Office of Legal Counsel’s long-held position about the president’s powers under the vacancies law.
Confirming a permanent director for the agency could take months. Mr. Cordray’s confirmation was delayed for two years by Republicans and the banking industry, two parties that objected to the agency’s creation and sought to limit the attempt at federal oversight.
If he takes office, Mr. Mulvaney will shoulder his new job as acting chief on top of his role as director of the Office of Management and Budget. It is unclear how he will manage both. Administration officials directed all questions about the mechanics of the situation to Mr. Mulvaney.
“We think he’ll show up Monday, go into the office and start working,” an administration official said of Mr. Mulvaney’s plans.
But it may not be that simple. The next move lies with Ms. English: She will need to decide whether to mount a legal challenge against Mr. Mulvaney for the bureau’s leadership, or to defer to him.
“It’s a very awkward legal scenario if they both show up literally at the same office,” said Andy Grewal, a law professor at the University of Iowa. “It seems like both sides are trying to engage in gamesmanship here.”