The information came from a partner of the US which had not given the US permission to share it with Russia, says the Washington Post.
It happened when Mr Trump met Sergei Lavrov in the Oval Office last week, says the paper.
But a senior security official has said the report is not true.
Media captionUS National Security Advisor H.R McMaster: "I was in the room, it didn't happen"
"The story that came out tonight, as reported, is false," National Security Advisor HR McMaster told reporters outside the White House.
"The president and 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."
The Trump campaign's alleged links to Moscow have dogged his presidency and are part of several investigations.
But the president has dismissed such allegations as "fake news".
During the election campaign, Mr Trump repeatedly criticised his Democratic opponent, Hillary Clinton, for how she handled sensitive material.
Russia: The scandal Trump can't shake
A knife in the back? Analysis by Anthony Zurcher, BBC News, Washington
The fallout from this story could be enormous and not just because there is a boundless trove of Republican quotes over the past year - directed at Mrs Clinton - about the utmost importance of protecting top-secret information.
There is the Russian connection, of course.
The FBI is currently investigating the Trump campaign for possible ties to Russian interests. Meetings with Russian Ambassador Sergei Kislyak featured prominently in the firing of National Security Advisor Michael Flynn and Attorney General Jeff Sessions' recusal on Russian investigation matters.
Then there is the question of whether US allies will be more reluctant to share sensitive intelligence information with the US, lest the president put sources at risk.
This will only stoke accusations by Trump critics that the president is undisciplined and inexperienced in the delicacies of foreign policy, where his shoot-from-the hip style presents an ongoing danger.
Finally, it is worth remembering the simmering feud Mr Trump has had with the US intelligence community. It took less than a week for this highly embarrassing story to leak. If the revelation was a knife twisted in the president's back, it is not hard to suspect where it came from.
What actually happened?
In a conversation with the Russian foreign minister and the Russian ambassador, Sergei Kislyak, in the Oval Office, the president revealed details that could lead to the exposure of a source of information, officials told the Washington Post and the New York Times.
The discussion was about an IS plot and he reportedly went "off-script". The intelligence disclosed came from a US ally, information considered too sensitive to share with other US allies, the papers report.
Others present realised the mistake and scrambled to "contain the damage" by informing the CIA and the National Security Agency, says the Post.
The meeting came a day after Mr Trump fired his FBI chief, James Comey, sparking criticism that he had done so because the FBI was investigating alleged Russian ties.
A Russian photographer was present for part of the meeting but US media were not allowed to attend.
The Trump-Comey firestorm explained
What has the reaction been?
The Senate's second-highest ranked Democrat, Dick Durbin, said Mr Trump's actions appeared to be "dangerous" and reckless".
The Republican head of the Senate Foreign Relations Committee, Bob Corker, said the story was "very, very troubling" if true.
"Obviously they're in a downward spiral right now and they've got to figure out a way to come to grips" with it, he told Bloomberg.
And the White House?
As well as Mr McMaster's short address to reporters, there was a statement issued by Secretary of State Rex Tillerson, which echoed the point that "the nature of specific threats were discussed, but they did not discuss sources, methods or military operations".
The Washington Post, which first broke the story, said this did not amount to a denial of their story.
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