Ever since F.B.I. investigators discovered in 2013 that a Russian spy was trying to recruit an American businessman named Carter Page, the bureau maintained an occasional interest in Mr. Page. So when he became a foreign policy adviser to the Trump campaign last year and gave a Russia-friendly speech at a prestigious Moscow institute, it soon caught the bureau’s attention.That trip last July was a catalyst for the F.B.I. investigation into connections between Russia and President Trump’s campaign, according to current and former law enforcement officials.It is unclear exactly what about Mr. Page’s visit caught the F.B.I.’s attention: meetings he had during his three days in Moscow, intercepted communications of Russian officials speaking about him, or something else.After Mr. Page, 45 — a Navy veteran and businessman who had lived in Moscow for three years — stepped down from the Trump campaign in September, the F.B.I. obtained a warrant from the Foreign Intelligence Surveillance Court allowing the authorities to monitor his communications on the suspicion that he was a Russian agent.From the Russia trip of the once-obscure Mr. Page grew a wide-ranging investigation, now accompanied by two congressional inquiries, that has cast a shadow over the early months of the Trump administration. At a House Intelligence Committee hearing last month, the F.B.I. director, James B. Comey, took the unusual step of publicly acknowledging the investigation of Russian interference in the election, which he said included possible links between Russia and Trump associates.