Since Mr. Mueller was appointed last May, Mr. Trump had taken a largely nonconfrontational approach to the investigation, providing tens of thousands of pages of emails, notes, memos and other documents as part of an effort to show he has nothing to hide and to hasten the end of the investigation.

As recently as December, Mr. Trump said he believed Mr. Mueller would treat him fairly. And Mr. Trump has repeatedly said in public and in private that he wanted to sit with Mr. Mueller for an interview. After the search warrant, Mr. Trump now is convinced that his initial belief that Mr. Mueller is simply out to destroy his presidency was correct.

Sarah Huckabee Sanders, the White House press secretary, criticized Mr. Mueller’s investigation on Wednesday for going beyond its mandate to look into Russia’s meddling in the 2016 election and into the ties between Mr. Trump’s campaign and Russia.

“The president certainly has been clear that he has very deep concern about the direction that the special counsel and other investigations have taken,” Ms. Sanders said in response to a question about a report that Mr. Trump came close to firing Mr. Mueller in December. “This investigation started off as Russian collusion, of which there was none.”

It is not clear what role, if any, Mr. Cohen played regarding the “Access Hollywood” tape, which was made public a month before the election on one of the more memorable days of the campaign. On that day, Oct. 7, the Obama administration called out Russia for meddling in the election and the first batch of emails from Hillary Clinton’s campaign were released.

But Mr. Cohen has acknowledged paying $130,000 to Ms. Clifford, who said she had a sexual encounter with Mr. Trump and signed a nondisclosure agreement promising not to discuss the matter. Mr. Cohen has insisted there was no relationship, but that he sought to keep a damaging story from emerging regardless.

Mr. Cohen also had a long relationship with David J. Pecker, the publisher of The National Enquirer, who is also friends with Mr. Trump and who engaged in the practice of “catch and kill” with negative stories, meaning women who made accusations of sexual relationships with the candidate received payments or contracts with the magazine.