The paper said the comments came from the White House's written account of the Oval Office meeting.
One official had read quotations to the Times and another had confirmed the broad outlines of the discussion.
The report quotes Mr Trump as calling ousted FBI director James Comey "crazy" and "a real nut job".
It says the President then told Russia's foreign minister and ambassador that he "faced great pressure because of Russia. That's taken off".
Mr Trump met with the Russians on 10 May - the day after Mr Comey was fired.
The developments were a blow to White House efforts to dampen down interest in the Russia investigation as Mr Trump and his staff boarded Air Force One for Saudi Arabia, the first stop on his first foreign trip as President.
In another development, The Washington Post is reporting that a current senior White House adviser is under scrutiny as part of an investigation into possible co-ordination between the Trump campaign and Russia.
The Post says the revelation comes as the investigation appears to be entering a more open and active phase, with investigators conducting interviews and using a grand jury to issue subpoenas.
Administration officials who have acknowledged contacts with Russian officials include Mr Trump's son-in-law Jared Kushner, attorney general Jeff Sessions and secretary of state Rex Tillerson.
Earlier this week, the Justice Department appointed former FBI director Robert Mueller to take over the federal investigation in an effort to re-establish independence from the White House.
On Friday, deputy attorney general Rod Rosenstein told Congress he stands by a memo he wrote bluntly criticising Mr Comey.
However, he made clear he did not intend for Mr Trump or other White House officials to use the document to justify firing Mr Comey, which is what they have done.
In meetings behind closed doors with politicians on Thursday and Friday, Mr Rosenstein said he wrote the memo after Mr Trump told him one day before the 9 May sacking that he wanted to dismiss Mr Comey.
Mr Rosenstein said that though he was personally fond of Mr Comey, he thought it was "appropriate" to seek a new leader.
The Justice Department issued the text of Mr Rosenstein's opening remarks for the briefings on Capitol Hill.
Source : https://news.sky.com/story/president-trump-told-russians-firing-nut-job-comey-relieved-pressure-10884848