Seeking deals in Russia became part of a broader strategy to expand the Trump brand worldwide. By the mid-2000s, Mr. Trump was transitioning to mostly licensing his name to hotel, condominium and commercial towers rather than building or investing in real estate himself. He discovered that his name was especially attractive in developing countries where the rising rich aspired to the type of ritzy glamour he personified.

While he nailed down ventures in the Philippines, India and elsewhere, closing deals in Russia proved challenging. In 2008, Donald Trump Jr. praised the opportunities in Russia, but also called it a “scary place” to do business because of corruption and legal complications.

Mr. Sater said that American hotel chains that had moved into Russia did so with straightforward agreements to manage hotels that other partners owned. Mr. Trump, by contrast, was pursuing developments that included residential or commercial offerings in which he would take a cut of sales, terms that Russians were reluctant to embrace.

Even so, Mr. Trump said his efforts put him in contact with powerful people there. “I called it my weekend in Moscow,” Mr. Trump said of his 2013 trip to Moscow during a September 2015 interview on “The Hugh Hewitt Show.” He added: “I was with the top-level people, both oligarchs and generals, and top of the government people. I can’t go further than that, but I will tell you that I met the top people, and the relationship was extraordinary.”

When asked about Mr. Trump’s claim that he had “stayed away” from Russia, Alan Garten, general counsel for the Trump Organization, said it was a fair characterization given that none of the development opportunities ever materialized. Mr. Trump’s interest in Russia, he said, was no different from his attraction to other emerging markets in which he investigated possible ventures. Mr. Garten did not respond to questions about whom Mr. Trump met with in Moscow in 2013 and what was discussed.

Stalking Deals

Ted Liebman, an architect based in New York, got the call in 1996. Mr. Trump and Liggett-Ducat, an American tobacco company that owned property in Moscow, wanted to build a high-end residential development near an old Russian Olympic stadium. As they prepared to meet with officials in Moscow, they needed sketches of the Trump tower they envisioned.

The architect scrambled to meet the request, handing over plans to Mr. Trump at his Manhattan office. “I hope we can do this,” Mr. Liebman recalled Mr. Trump telling him.