Mr. Cordray’s five-year term was scheduled to end in July 2018. He was widely expected to step down early to run for governor in Ohio, his home state.

“Together we have made a real and lasting difference,” Mr. Cordray wrote in an email to the agency’s employees. “I am confident that you will continue to move forward, nurture this institution we have built together, and maintain its essential value to the American public. And I trust that new leadership will see that value also and work to preserve it.”

Mr. Cordray did not discuss his future plans in his email. A bureau spokesman declined to comment.

In the months since Mr. Trump was elected, Mr. Cordray unleashed a fusillade of regulations — with mixed success. The agency adopted a new rule in July that would have allowed consumers to band together in class actions against financial institutions. Last month, Congress overturned the rule, with Vice President Mike Pence breaking a 50-50 tie in the Senate.

Other rules, including new protections on prepaid debit cards, withstood challenges. The agency recently imposed restrictions that will sharply curtail the payday lending market.

By installing his own director, Mr. Trump will be able to reshape the agency to look more like other financial regulators, many of which are now run by former industry executives. He picked Randal K. Quarles, a former private equity executive and banking industry lawyer to fill one of the top jobs at the Federal Reserve, and Steven Mnuchin, a former investment banker at Goldman Sachs, heads the Treasury Department.

One of the names being floated for the consumer bureau’s leadership post is Todd J. Zywicki, a conservative law professor at George Mason University, according to two people familiar with the White House’s discussions. They requested anonymity because the discussions are not public.

Democrats pledged to fight any nominee they perceive to be too cozy with the financial industry.

“The White House has said it wants to stand up for the middle class,” said Senator Sherrod Brown, an Ohio Democrat. “If that’s true, the president must nominate a successor who will put working people ahead of Wall Street.”