The allegations were in a two-page synopsis appended to a report presented by U.S. intelligence officials to Trump and President Barack Obama on Russian interference in the 2016 election, CNN said, citing multiple U.S. officials with direct knowledge of the briefings.
The allegations came, in part, from memos compiled by a former British intelligence operative, whose past work U.S. intelligence officials consider credible, CNN reported.
It said the FBI is investigating the credibility and accuracy of these allegations, which are based primarily on information from Russian sources, but has not confirmed many essential details in the memos about Trump.
The FBI declined comment.
Allegations that Russia attempted to compromise New York real estate businessman Trump have been circulating in Washington for months, and they were presented to U.S. officials last year.
Reuters has not been able to corroborate the allegations or confirm the existence of the two-page synopsis reported by CNN.
The warning to Trump, however, coincides with growing concern about what Director of National Intelligence James Clapper has called a "multifaceted" Russian influence and espionage campaign in Europe and the United States.
In addition to hacking computer networks and spreading propaganda and fake news, it includes efforts to cultivate business and political leaders and find compromising personal, financial or other information on persons of interest, U.S. intelligence officials said.
The classified briefings last week were presented by Clapper, Federal Bureau of Investigation Director James Comey, Central Intelligence Agency Director John Brennan and National Security Agency Director Mike Rogers.
One reason the U.S. intelligence chiefs took the extraordinary step of including the synopsis in the briefing documents was to make Trump aware that allegations about him are circulating among intelligence agencies, senior members of Congress and government officials, CNN said.
An unclassified intelligence report released on Friday concluded that Russian President Vladimir Putin ordered an effort to help Trump's electoral chances by discrediting Democrat Hillary Clinton in the 2016 presidential campaign.
The report said U.S. intelligence agencies have concluded that as part of the effort Russia's military intelligence agency, the GRU, used intermediaries such as WikiLeaks and others to release emails it hacked from the Democratic National Committee and top Democrats.