There was no indication, in reporting by CNN and other media, of who might be charged or what crimes might be alleged in the ongoing inquiry led by former FBI chief Robert Mueller.
But Trump, in a rapid burst of tweets early Sunday, again denounced the investigation as a "witch hunt" and repeated his denials of any collusion with Russia.
Mueller's team has remained publicly mum about reports that a first arrest could come as early as Monday. He is empowered to pursue not only Russian interference but any other crimes his large team of prosecutors should uncover.
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But Chris Christie, a Republican governor close to Trump, said Sunday on ABC that "the important thing about today for the American people to know is the president is not under investigation. And no one has told him that he is."
It was not clear the New Jersey governor would know whether Trump is being investigated; he may have been referring to earlier comments by former FBI chief James Comey.
But Christie told CNN that anyone who has been advised by Mueller's office that they are a target of the inquiry "should be concerned."
'I cannot answer'
Typically, such a wide-ranging investigation would first target lower-level people while building a case against those higher up. Sometimes early indictments are used to pressure potential witnesses into turning against others.
Representative Adam Schiff of California, the ranking Democrat on the House intelligence committee, demurred Sunday when asked whether Trump was under investigation. "I can't answer that one way or the other," he told ABC.
But he mentioned two possible targets on whom much speculation has focused: former national security adviser Michael Flynn and former Trump campaign director Paul Manafort, both of them once involved in undeclared lobbying for foreign interests.
Buzzfeed reported Sunday the FBI is probing a series of wire transfers linked to Manafort that saw offshore companies move over $3 million between 2012 and 2013.
Flagged as suspicious by US financial institutions, some of the 13 transfers Buzzfeed obtained details of involved wires of large sums from Ukraine.
According to the news site, federal law enforcement was aware of such transfers as far back as 2012, when they began investigating whether Manafort committed tax fraud or helped the Ukrainian regime -- close to Vladimir Putin -- launder money.
White House lawyer Ty Cobb told the New York Times that Trump felt confident that neither Flynn nor Manafort had damaging information to offer prosecutors.
"The president has no concerns in terms of any impact, as to what happens to them, on his campaign or on the White House," Cobb said in an interview published Saturday.
Cobb also asserted Sunday that Trump's latest tweets were "unrelated to the activities of the Special Counsel, with whom he continues to cooperate," news media reported.
As the Mueller investigation nears a dramatic new phase, Republican officials and conservative media have stepped up their attacks on Democrats, above all on Trump's rival in last year's election, Hillary Clinton -- attacks that Democrats dismiss as blatant attempts to divert attention.
'So much GUILT'
Trump, in his tweets Sunday, again complained of Clinton's handling of emails while secretary of state, of Democratic Party funding of what he said was a "fake" dossier on Trump's background, and of a US sale during the Obama administration of uranium rights to Russia.
"There is so much GUILT by Democrats/Clinton, and now the facts are pouring out. DO SOMETHING!" Trump tweeted.
Trump's mention of the "fake" dossier appeared to refer to revelations that the Clinton campaign and Democratic National Committee had funded part of the research by a former British intelligence agent into possible links between Trump, his collaborators and Russia.
In the uranium case, Russian nuclear energy agency Rosatom sought in 2010 to buy a share in Toronto-based Uranium One, which has mining stakes in the United States. A panel of nine US government agencies, including the State Department, approved the sale, though Clinton says she was "not personally involved."
As Mueller's inquiry advances, there have been calls from a few Republicans -- and from the conservative editorial board of the Wall Street Journal -- for him to resign. Christie cautioned on Sunday that the former FBI chief should be "very, very careful" about proceeding with transparency and credibility.
Democrats meantime have warned that if Trump were to fire Mueller -- or issue preemptive pardons to anyone caught in his net -- it would be crossing a dangerous line, potentially sparking a constitutional crisis.
