During the meeting in the Oval Office, Mr Trump is reported to have given details about an IS threat related to the use of laptop computers in aircraft.
The newspaper says the information the President relayed had been provided by a partner through an intelligence-sharing arrangement considered so sensitive that details had been withheld from allies and tightly restricted, even within the US government.
A national Security adviser to the President, HR McMaster, was at the meeting.
He's confirmed the subject of threats to civil aviation was discussed, but says it was only in general terms.
"There's nothing that the President takes more seriously than the security of the American people. The story that came out tonight as reported is false. The President and the Foreign Minister reviewed a range of common threats to our two countries, including threats to civil aviation. At no time, at no time, were intelligence sources or methods discussed. And the President did not disclose any military operations that were not already publicly known. Two other senior officials who were present - including the Secretary of State - remember the meeting the same way and have said so. Their on-the-record accounts should outweigh those of anonymous sources. I was in the room. It didn't happen."
Secretary of State Rex Tillerson has released a statement supporting General McMaster.
He says a number of subjects were discussed with the Russians, among which were common efforts and threats regarding counter-terrorism.
Mr Tillerson says the nature of specific threats was discussed, but they did not discuss sources, methods or military operations.
But the journalist who broke the story, Greg Miller, says his story makes no mention of sources or methods being disclosed.
He says the US officials have not addressed whether Mr Trump disclosed information drawn from sensitive sources.
Greg Miller says the article alleges a specific IS plot was discussed, from which sources and methods may be determined.
"Nor do any of these White House officials who are denouncing this story nor have any of them offered any explanation why, if this was all so aboveboard and not problematic in any way, why did the National Security Council coming out of this meeting feel it was necessary to contact the CIA director and the director of the National Security Agency to give them a heads-up on what Trump had just told the Russians?"
Constitutional lawyer Alan Dershowitz says the allegations are extremely grave.
"This is the most serious charge ever made against a sitting president. Let's not minimise it. Comey is in the waste-basket of history, everything else is off the table. This is the most serious charge ever made against a sitting president of the United States. Let's not underestimate it."
Mr Dershowitz has clarified that while the accusation is serious, it's not criminal and it's not impeachable.
Democratic Senator Patrick Leahy says the Washington Post report is deeply troubling.
"I don't know whether the article is accurate or not. They've had a lot of very accurate articles but it would be almost inconceivable that any president would allow something of that nature out, anything that would disclose sources and method, those are things that we protect."
