Investigators have also been looking for any evidence of possible financial crimes among Trump associates
It's being reported that the special counsel overseeing the investigation into Russia’s role in the 2016 election is examining whether President Trump attempted to obstruct justice.
The move by special counsel Robert S. Mueller III to investigate Trump’s conduct marks a major turning point in the nearly year-old FBI investigation, which until recently focused on Russian interference during the presidential campaign and on whether there was any coordination between the Trump campaign and the Kremlin.
Investigators have also been looking for any evidence of possible financial crimes among Trump associates, officials said.
Former FBI director James Comey told a Senate hearing last week he thought he had been dismissed "because of the Russia investigation".
He also testified he had told the President in January he was not personally under investigation.
That changed after Mr Comey was sacked last month, claim five officials who spoke anonymously to the Washington Post.
The paper says that Mr Mueller could interview key figures about possible obstruction by the President as early as next week.
They are said to be National Security Agency head Mike Rogers, his former deputy Richard Ledgett and national intelligence director Daniel Coats.
A spokesman for President Trump's lawyer responded to the Post's report by saying: "The FBI leak of information regarding the president is outrageous, inexcusable and illegal."
Robert Mueller - a former head of the FBI himself - was appointed in May as special counsel to investigate the alleged Trump-Russia links.
Four other US Congress committees are also investigating.
However, it appears the scope of Mr Mueller's inquiry is widening and he has the freedom to look at anything he believes relevant.
Recently fired FBI boss James Comey testified last week that he thought Mr Trump had got rid of him because of the bureau's own inquiry.
He told the Senate Intelligence Committee the President had asked him to drop an investigation into former national security adviser Michael Flynn, and to "lift the cloud" of the Russia investigation.
Mr Flynn resigned in February after it was revealed he had misled Vice-President Mike Pence over his contact with the Russian ambassador.
Mr Comey would not answer questions on whether the President's alleged request amounted to obstruction of justice, but said he was "stunned by the conversation".