In October, the bank said it would offer refunds to home buyers who were improperly charged fees in order to obtain low-rate mortgages. It was reviewing nearly $100 million in fees to 110,000 customers that were potentially improper.

Mr. Trump’s tweet may have been referencing a Reuters report published on Thursday that said Mr. Mulvaney was reviewing the penalties for Wells Fargo’s mortgage abuses. The report was based on information from three anonymous sources.

This year, the consumer bureau started an investigation into the mortgage lending practices at Wells Fargo. The investigation, according to two people familiar with it, centered on whether the bank incorrectly charged customers who wanted to lock in lower mortgage rates.

The inquiry was delayed amid a broader shake-up at the agency, according to the two people. The turmoil began last month when Mr. Trump appointed Mr. Mulvaney as the acting director of the agency after Richard Cordray resigned.

On his way out the door, Mr. Cordray named Leandra English as his acting deputy director and presumed acting director. After Mr. Trump tapped Mr. Mulvaney to the role, it created the unusual scenario of two people laying claim to the same post at the helm of a federal agency, with lawyers debating whether the Dodd-Frank Act, which created the consumer bureau, or the Federal Vacancies Reform Act provided the authoritative answer to who has the authority to install an acting director.