It was unclear what specific objections Mr. Mueller raised in his letter, though a Justice Department spokeswoman said on Tuesday evening that he “expressed a frustration over the lack of context” in Mr. Barr’s presentation of his findings on obstruction of justice. Mr. Barr defended his descriptions of the investigation’s conclusions in conversations with Mr. Mueller over the days after he sent the letter, according to two people with knowledge of their discussions.

Mr. Barr, who was scheduled to testify on Wednesday before the Senate Judiciary Committee about the investigation, has said publicly that he disagrees with some of the legal reasoning in the Mueller report. Senior Democratic lawmakers have invited Mr. Mueller to testify in the coming weeks but have been unable to secure a date for his testimony.

[Update: Trump objects to Mueller testifying before Congress.]

A central issue in the simmering dispute is how the public’s understanding of the Mueller report has been shaped since the special counsel ended his investigation and delivered his 448-page report on March 22 to the attorney general, his boss and longtime friend. The four-page letter that Mr. Barr sent to Congress two days later gave little detail about the special counsel’s findings and created the impression that Mr. Mueller’s team found no wrongdoing, allowing Mr. Trump to declare he had been exonerated.

But when Mr. Mueller’s report was released on April 18, it painted a far more damning picture of the president and showed that Mr. Mueller believed that significant evidence existed that Mr. Trump obstructed justice.

“The special counsel emphasized that nothing in the attorney general’s March 24 letter was inaccurate or misleading,” a Justice Department spokeswoman, Kerri Kupec, said in response to a request for comment made on Tuesday afternoon. A spokesman for the special counsel’s office declined to comment.