Senator Richard Blumenthal, Democrat of Connecticut, said Mr. Mulvaney was “hardly the kind of peacemaker” needed to smooth relations between the White House and Democrats on Capitol Hill.

“What he did at the Consumer Financial Protection Bureau was destroy an agency and undermine its mission,” Mr. Blumenthal said in an interview. “He seems to be a Trump surrogate with a clear agenda, and my fear is he will exacerbate divisions at a time when they need to be bridged.”

Last year, Mr. Trump asked him to oversee the consumer bureau, an agency the president was bent on diminishing. That led to a messy public spat and court hearing as the Trump administration wrested control over the financial watchdog from an Obama-era holdover.

Mr. Mulvaney quickly went to work on the president’s mission. He seemed to relish his role at the agency, where he tried to vastly curtail its activities, including virtually freezing new investigations and changing its well-known name in an attempt to unbrand it.

When he first took over, he promised to split his time between the budget office and the consumer bureau, which is within walking distance of the White House. But as the months dragged on, he spent less time at the bureau and toward the end of his tenure there employees reported seeing him no more than once or twice a week.

Mr. Mulvaney was one of the few prospects for the chief of staff job who was seen as openly campaigning for it over most of the year. At one point, he told the president he was right for it because he was the only person in the administration leading a department that was not mired in scandal, according to a person familiar with the discussion.

Sarah Huckabee Sanders, the White House press secretary, said Mr. Mulvaney was not resigning from his job at the budget office, but would spend all of his time as chief of staff. He will turn over running the department to Russ Vought, the office’s deputy director, at a somewhat precarious time. The budget deficit widened to $305 billion in October and November, compared with $202 billion from the same period last year, the Treasury Department reported on Thursday.