The goal would probably never have been to make Mr. Trump a knowing agent of Russia, but to make him a source who might provide information to friendly Russian contacts. But if Mr. Putin and his agents wanted to entangle Mr. Trump using business deals, they did not do it very successfully. Mr. Trump has said he has no major properties there, though one of his sons said at a real estate conference in 2008 that “a lot of money” was “pouring in from Russia.”

The second Russian operation described was recent: a series of contacts with Mr. Trump’s representatives during the campaign, in part to discuss the hacking of the Democratic National Committee and Mrs. Clinton’s campaign chairman, John D. Podesta. According to Mr. Steele’s sources, it involved, among other things, a late-summer meeting in Prague between Michael Cohen, a lawyer for Mr. Trump, and Oleg Solodukhin, a Russian official who works for Rossotrudnichestvo, an organization that promotes Russia’s interests abroad.

By all accounts, Mr. Steele has an excellent reputation with American and British intelligence colleagues and had done work for the F.B.I. on the investigation of bribery at FIFA, soccer’s global governing body. Colleagues say he was acutely aware of the danger he and his associates were being fed Russian disinformation. Russian intelligence had mounted a complex hacking operation to damage Mrs. Clinton, and a similar operation against Mr. Trump was possible.

But much of what he was told, and passed on to Fusion GPS, was very difficult to check. And some of the claims that can be checked seem problematic. Mr. Cohen, for instance, said on Twitter on Tuesday night that he has never been in Prague; Mr. Solodukhin, his purported Russian contact, denied in a telephone interview that he had ever met Mr. Cohen or anyone associated with Mr. Trump. The president-elect on Wednesday cited news reports that a different Michael Cohen with no Trump ties may have visited Prague and that the two Cohens might have been mixed up in Mr. Steele’s reports.

But word of a dossier had begun to spread through political circles. Rick Wilson, a Republican political operative who was working for a super PAC supporting Marco Rubio, said he heard about it in July, when an investigative reporter for a major news network called him to ask what he knew.

By early fall, some of Mr. Steele’s memos had been given to the F.B.I., which was already investigating Mr. Trump’s Russian ties, and to journalists. An MI6 official, whose job does not permit him to be quoted by name, said that in late summer or early fall, Mr. Steele also passed the reports he had prepared on Mr. Trump and Russia to British intelligence. Mr. Steele was concerned about what he was hearing about Mr. Trump, and he thought that the information should not be solely in the hands of people looking to win a political contest.