For much of the summer, the F.B.I. pursued a widening investigation into a Russian role in the American presidential campaign. Agents scrutinized advisers close to Donald J. Trump, looked for financial connections with Russian financial figures, searched for those involved in hacking the computers of Democrats, and even chased a lead — which they ultimately came to doubt — about a possible secret channel of email communication from the Trump Organization to a Russian bank.

Law enforcement officials say that none of the investigations so far have found any link between Mr. Trump and the Russian government. And even the hacking into Democratic emails, F.B.I. and intelligence officials now believe, was aimed at disrupting the presidential election rather than electing Mr. Trump…

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.