The effort to sow doubt about Mr. Mueller’s team and the department that appointed him has gained steam since early December after Michael T. Flynn, Mr. Trump’s former national security adviser, pleaded guilty to lying to the F.B.I. and cut a deal to cooperate with Mr. Mueller’s inquiry.

The following day, The Times reported that Mr. Mueller had removed a top F.B.I. agent last summer over text messages expressing anti-Trump political views that he had exchanged with an agency lawyer. The news — and the Justice Department’s release days later of many of the text messages — provided Republicans ammunition of a sort they had long lacked: Senior officials involved in both the investigation of the Trump campaign and of Hillary Clinton’s use of a private email server railing against the president in private.

Since then, Republicans who control key House committees have called top Justice and F.B.I. officials to Capitol Hill for hourslong interviews in public and behind closed doors about the handling of the Clinton and Trump investigations. They have accused one of Mr. Mueller’s top deputies of anti-Trump bias based on an email sent early last year praising the acting attorney general for her decision not to defend Mr. Trump’s first travel ban in court.

And they have pointed to the actions of another senior Justice Department official, Bruce Ohr, as a predicate for the appointment of a second special counsel to investigate political partisanship in the department’s handling of the Trump-Russia investigation and its decision not to charge Mrs. Clinton in the email case.

“The public trust in this whole thing is gone,” Representative Jim Jordan, an Ohio Republican who has helped lead the charge, said during a tense Dec. 13 hearing with the Deputy Attorney General Rod J. Rosenstein.

Mr. Trump himself has alternated between playing coach and cheerleader, repeatedly painting the Mueller and congressional investigations as a partisan attack and top F.B.I. officials as Clinton loyalists. In the Thursday interview, he said Mr. Mueller’s investigation made the country “look very bad.”