Then, after Mr. Parnas secured Mr. Giuliani’s help for one of his business ventures in a half-million-dollar deal, Mr. Giuliani saw that they had something else to offer him: the ability to make connections in Ukraine. And so, they landed at the fulcrum of an international effort that could both aid the president and anchor their future fortunes.

Mr. Parnas, now 47, was a perennial pitchman who exuded sincerity, an aspiring entrepreneur with a history of debts and aborted businesses. He tooled around in a Ferrari and wooed new sources of capital with boasts of ties to important people.

Mr. Fruman, a 53-year-old who spoke a mix of Russian and choppy English, had more direct links to Ukraine: He invested in a sprawl of businesses there that were once worth millions of dollars — nightspots, a beach club, a dealership selling Jaguars that was part-owned by a local politician — as well as a New York-based import-export company. But in a sign that his fortunes had started to change, Mr. Fruman funded the men’s entree into Republican circles by refinancing a condominium in Sunny Isles, a South Florida enclave of Russian émigrés.

Together, they became ubiquitous in the Republican donor set, known simply as “Lev and Igor,” a duo seemingly ripped from a movie script.

This account is based on interviews with more than three dozen people in the United States and Ukraine and a review of photographs, credit card transactions, internal memos, and American and Ukrainian court documents and corporate filings.

While Mr. Parnas’s Instagram posts show appearances at Republican fund-raisers and Mr. Giuliani’s birthday party at Yankee Stadium, it was in private meetings at BLT Prime, the steakhouse at the Trump International Hotel in Washington, where he and Mr. Fruman became unlikely agents of the Ukrainian initiative.

Mr. Parnas did most of the talking, but Mr. Fruman’s Ukrainian ties were critical to their efforts. Two people familiar with the matter said Mr. Fruman helped arrange an initial meeting between Mr. Giuliani and a Ukrainian prosecutor who claimed to have damaging information about targets of Mr. Trump, laying part of the foundation for their mission.