NEW YORK ­— President Trump has named his budget director as acting director of the Consumer Financial Protection Bureau, moving to take control of the agency after its departing leader had taken steps to install his own choice for acting chief.

The agency — which was created in response to the financial meltdown and which Republicans have tried to kill from the start — now has dueling directors, and there is little sense of who will be in charge Monday morning.

The bureaucratic standoff began Friday afternoon when Richard Cordray, the Obama-appointed leader of the bureau, abruptly announced he would leave the job at the close of business, a week earlier than anticipated.

He followed up with a letter naming his chief of staff, Leandra English, as the agency’s deputy director.

The bureau was proposed in 2007 by Elizabeth Warren, then a Harvard law professor, but she was passed over to lead the agency after Obama administration officials became concerned that she would not be able to overcome resistance from Republicans during the confirmation process.

On Friday, Warren defended Cordray’s decision on Facebook: “President Trump can’t override that. He can nominate the next CFPB Director — but until that nominee is confirmed by the Senate, Leandra English is the Acting Director under the Dodd-Frank Act.’’

In naming English as deputy director Friday, Cordray said explicitly that the appointment would make her the agency’s acting director. The move was seen as an effort to delay Trump from appointing his own director, whose confirmation could take months.

The White House retaliated, saying that the budget director, Mick Mulvaney, who once characterized the consumer protection bureau as a “sad, sick joke,’’ would be running the agency. He would also keep his current job as head of the Office of Management and Budget.

The White House officials said Saturday that the president has the authority to make the provisional appointment under the Federal Vacancies Reform Act of 1998.

Mulvaney said he would assume the additional role until a permanent successor was found.

“I believe Americans deserve a CFPB that seeks to protect them while ensuring free and fair markets for all consumers,’’ he said. “Financial services are the engine of American democratic capitalism, and we need to let it work.’’

Cordray named English as deputy director in a letter to the consumer protection agency’s staff.

Under the 2010 Dodd-Frank Act, which established the regulatory agency, the deputy director is to serve as acting director in the absence of a permanent leader, Cordray said.

The conflicting appointments were a fitting development for an agency under constant attack from Republican leaders, and the confusion leaves supporters wondering about the agency’s future with Trump in the White House and Republicans in control of both houses of Congress.

Cordray, a former attorney general of Ohio, had been initially selected by Warren to work for her as the agency’s enforcement director before Obama tapped him to be director.

For two years, Republicans prevented the confirmation of a director. The agency’s creation was also largely opposed by the banking industry, which sought to prevent Cordray’s confirmation.

In July 2013, the Senate finally agreed to allow the confirmation of Cordray, cementing a new era of expansive federal oversight of companies that lend money to consumers.

The fight over the establishment of the protection bureau made Warren a political celebrity and helped her win a bid for Congress. Later, as a senator from Massachusetts, Warren has continued to champion the bureau, which has emerged as an aggressive consumer watchdog with broad power to combat financial abuses.

Consumer groups on Friday praised Cordray, whose move appeared to be a final attempt to keep the bureau in hands that would preserve the legacy he helped create as its first director.

“Fortunately, the statute creating the CFPB says that the agency’s deputy director serves as acting director until a new director has been nominated by the president and confirmed by the Senate,’’ said Lisa Donner, executive director of Americans for Financial Reform.

“Mulvaney has said he is opposed to the very existence of the CFPB, and as a member of Congress he voted in favor of Wall Street banks and predatory lenders — his largest donors — again and again,’’ she said.

The appointment of Mulvaney, who as a Republican congressman from South Carolina was a cosponsor of legislation to shut down the consumer bureau, had been widely anticipated. The White House said in a statement Friday that Trump looked forward to seeing Mulvaney take a “common sense approach’’ to leading the bureau’s staff.

In a separate development Saturday, Trump said Time magazine was contemplating naming him the Person of the Year for a second year in a row but that he pulled himself out of the running because he didn’t want to participate in an interview and photo shoot.

‘‘I said probably is no good and took a pass. Thanks anyway!’’ Trump said in a Twitter message Friday evening.

Later Friday evening, Time disputed the president’s claim, tweeting that he ‘‘is incorrect about how we choose Person of the Year.’’ The magazine said it doesn’t comment on its choice until publication, the Associated Press reported.