The letter came days after the Justice Department took the unusual step of releasing to the news media anti-Trump text messages that an agent overseeing the investigation had sent to a colleague. Although Mr. Mueller had moved quickly this past summer to remove the agent from the inquiry, Republicans seized on the disclosure to criticize Mr. Mueller and the F.B.I.

One Trump adviser, Kellyanne Conway, said it was evidence that “the fix was in against Donald Trump from the beginning.”

White House officials sought to play down the significance of the letter about the transition emails, insisting it was an issue for the transition team, not the West Wing. They said the president had not changed his approach to cooperating with the special counsel, and that he had not discussed dismissing Mr. Mueller.

“We have been cooperative and transparent with the special counsel’s office and will continue to be — we look forward to an expeditious conclusion to this matter,” said Jay Sekulow, a lawyer for Mr. Trump.

Still, the letter to Congress from Mr. Langhofer sounded a discordant note. His assertion that the emails were privileged and should have been shielded stood in contrast to the stance of White House officials, who said that Mr. Trump’s lawyers had not invoked any such privilege on any White House documents that Mr. Mueller had requested.

Among the materials obtained from the transition team by Mr. Mueller were emails, laptops and cellphones for nine members who worked on national security and policy matters, according to the letter. Mr. Mueller’s investigators have used the documents during interviews with transition team officials when questioning them about calls between Mr. Trump’s former national security adviser, Michael T. Flynn, and the Russian ambassador in which they discussed American sanctions.

One of the emails shows that several transition officials were aware that Mr. Flynn was going to be speaking with the ambassador on Dec. 29 after the Obama administration had imposed new sanctions on Russia for its election meddling. Mr. Flynn pleaded guilty this month to lying to the F.B.I. about his interactions with the Russian official.