Mr. Nader tempted the fund-raiser, Mr. Broidy, with the prospect of more than $1 billion in contracts for his private security company, Circinus, and he helped deliver deals worth more than $200 million with the United Arab Emirates. He also flattered Mr. Broidy about “how well you handle Chairman,” a reference to Mr. Trump, and repeated to his well-connected friend that he told the effective rulers of both Saudi Arabia and the U.A.E. about “the Pivotal Indispensable Magical Role you are playing to help them.”

Mr. Nader’s cultivation of Mr. Broidy, laid out in documents provided to The New York Times, provides a case study in the way two Persian Gulf monarchies have sought to gain influence inside the Trump White House. Mr. Nader has been granted immunity in a deal for his cooperation with the special counsel, Robert S. Mueller III, according to people familiar with the matter, and his relationship with Mr. Broidy may also offer clues to the direction of that inquiry.

Mr. Nader has now been called back from abroad to provide additional testimony, one person familiar with the matter said this week. Mr. Mueller’s investigators have already asked witnesses about Mr. Nader’s contacts with top Trump administration officials and about his possible role in funneling Emirati money to Mr. Trump’s political efforts, a sign that the investigation has broadened to examine the role of foreign money in the Trump administration.

The documents contain evidence not previously reported that Mr. Nader also held himself out as intermediary for Saudi Arabia’s crown prince, Mohammed bin Salman, who met with Mr. Trump on Tuesday in the Oval Office at the beginning of a tour of the United States to meet with political and business leaders.

A lawyer for Mr. Nader declined to comment. Two people close to Mr. Broidy said he had not been contacted by the special counsel’s investigators. In a statement, Mr. Broidy said that his efforts “aimed to strengthen the national security of the United States, in full coordination with the U.S. government.” He added, “I have always believed strongly in countering both Iran and Islamic extremism, and in working closely with our friends in the Arab world in order to do so.”