The idea that Mr. Trump should order Mr. Gulen deported was fiercely opposed inside the White House, where officials saw the issue as a matter to be handled by the Justice Department, not a political decision. Ultimately, that was what happened. The Justice Department, which was led at the time by Attorney General Jeff Sessions, did not see merit in deporting Mr. Gulen, said one former official familiar with the matter.

Throughout 2017, before Mr. Giuliani began representing Mr. Trump as his personal lawyer, he appeared at the White House to discuss a number of issues related to Turkey, according to two former administration officials. At one point, officials tried to divert Mr. Giuliani’s access to the president so that he was raising his issues with the president’s senior advisers instead of Mr. Trump directly.

Also on Tuesday, Mr. Giuliani’s lawyer, Jon A. Sale, said in a letter to the House Intelligence Committee that he would not hand over documents to the Democrat-led impeachment inquiry because the information requested was “beyond the scope of legitimate inquiry” and a violation of attorney-client and executive privilege.

In a confrontational tweet that echoed the president’s condemnation of the investigations, Mr. Giuliani said he “will not participate in an illegitimate, unconstitutional, and baseless ‘impeachment inquiry.’”

Mr. Giuliani appeared to reject the idea that his decision to defy the subpoena would place him in any legal jeopardy. He said in his tweet that Mr. Sale would no longer be representing him on impeachment matters.

“At this time, I do not need a lawyer,” Mr. Giuliani wrote.

A spokeswoman for the committee did not immediately respond to a request for comment about Mr. Giuliani’s letter.

Mr. Giuliani has emerged as the central character in the monthslong effort by Mr. Trump and officials in his government to get Ukraine’s president to begin an investigation into former Vice President Joseph R. Biden Jr., a leading Democratic candidate for president, and his son Hunter Biden.