The Washington Post, quoting an unnamed person close to Mr Trump, says investigation lead Robert Mueller is likely to interview him within weeks.
Mr Trump’s legal team has not confirmed the reports.
Mr Mueller is investigating possible collusion between Trump’s 2016 campaign and Russia in the US elections.
Tensions between Mr Mueller, the special counsel appointed to look into alleged Russian interference, and the president have risen since the investigation led to charges against several former members of Mr Trump’s campaign team.
Mr Trump’s administration denies working with Russia on the election, and the president has labelled the investigation “a witch hunt”.
“This is moving faster than anyone really realises,” the newspaper quotes a person within the president’s circles, who spoke on condition of anonymity.
Mr Trump’s lawyers are reluctant to allow him to sit down for open-ended questioning and are discussing whether to allow him to provide written answers to some of the questions, The Washington Post and NBC News report.
According to NBC, which cited three people close to the story, the talks are “preliminary and ongoing”.
President Trump’s legal team did not confirm the reports, but told both media outlets: “The White House is continuing its full co-operation with the OSC [Office of Special Counsel Robert Mueller] in order to facilitate the earliest possible resolution.”
A spokesman for Mr Mueller also declined to comment.
The Post, quoting an unnamed source, says the two sides planned to meet again to continue discussions on terms and substance of any interview.
Aside from the Russia angle, special counsel Robert Mueller may also look into whether the president and his inner circle sought to obstruct justice while in office following the firing of FBI director James Comey.
Mr Trump in a later TV interview connected his sacking of Mr Comey with a prior investigation into Russia and the 2016 election led by the then-FBI boss.
Mr Comey testified that in a private meeting prior to his dismissal, the president had asked him to show leniency in an inquiry into his top aide, Michael Flynn.
President Trump has repeatedly denied accusations he pressed Mr Comey to drop his inquiries.
Robert Mueller was appointed by the justice department as special counsel shortly after Mr Comey’s dismissal.
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Sitting presidents have been interviewed by prosecutors in the past, most notably when Democrat President Bill Clinton testified before a grand jury in 1998 over his sexual relationship with White House intern Monica Lewinsky.
Justice department officials quoted by NBC say Mr Mueller is unlikely to pass up on an opportunity to interview the president directly.
Two of President Trump’s former aides, including Mr Flynn, have already been implicated in Mr Mueller’s investigation and are acting as witnesses for the probe.
Last month Mr Flynn, Mr Trump’s short-lived national security adviser, became the most senior official to be charged after admitting to making false statements to the FBI about meetings with Russia’s ambassador.
The US intelligence community has already concluded that Moscow tried to sway the presidential election in favour of Mr Trump, though Russia denies this.