F.B.I. officials declined to comment on Monday. Intelligence officials have said in interviews over the last six weeks that apparent connections between some of Mr. Trump’s aides and Moscow originally compelled them to open a broad investigation into possible links between the Russian government and the Republican presidential candidate. Still, they have said that Mr. Trump himself has not become a target. And no evidence has emerged that would link him or anyone else in his business or political circle directly to Russia’s election operations.

At least one part of the investigation has involved Paul Manafort, Mr. Trump’s campaign chairman for much of the year. Mr. Manafort, a veteran Republican political strategist, has had extensive business ties in Russia and other former Soviet states, especially Ukraine, where he served as an adviser to that country’s ousted president, Viktor F. Yanukovych.

But the focus in that case was on Mr. Manafort’s ties with a kleptocratic government in Ukraine — and whether he had declared the income in the United States — and not necessarily on any Russian influence over Mr. Trump’s campaign, one official said.

In classified sessions in August and September, intelligence officials also briefed congressional leaders on the possibility of financial ties between Russians and people connected to Mr. Trump. They focused particular attention on what cyberexperts said appeared to be a mysterious computer back channel between the Trump Organization and the Alfa Bank, which is one of Russia’s biggest banks and whose owners have longstanding ties to Mr. Putin.

F.B.I. officials spent weeks examining computer data showing an odd stream of activity to a Trump Organization server and Alfa Bank. Computer logs obtained by The New York Times show that two servers at Alfa Bank sent more than 2,700 “look-up” messages — a first step for one system’s computers to talk to another — to a Trump-connected server beginning in the spring. But the F.B.I. ultimately concluded that there could be an innocuous explanation, like a marketing email or spam, for the computer contacts.

The most serious part of the F.B.I.’s investigation has focused on the computer hacks that the Obama administration now formally blames on Russia. That investigation also involves numerous officials from the intelligence agencies. Investigators, the officials said, have become increasingly confident, based on the evidence they have uncovered, that Russia’s direct goal is not to support the election of Mr. Trump, as many Democrats have asserted, but rather to disrupt the integrity of the political system and undermine America’s standing in the world more broadly.

The hacking, they said, reflected an intensification of spy-versus-spy operations that never entirely abated after the Cold War but that have become more aggressive in recent years as relations with Mr. Putin’s Russia have soured.