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The White House raised the prospect of a staff purge over the disclosure, saying in a statement that it would be a "fireable offense and likely illegal" to give Trump's briefing papers to the news media.
Chief of staff John Kelly was "frustrated and deeply disappointed" by the leak, a White House official told The Hill. The official refused to say whether the White House has launched a formal investigation into the incident.
Only a small circle of staff members would have had access to such sensitive briefing documents, according to former White House officials, indicating that the leak may have some from somebody close to the president.
The incident raised concerns among Trump allies that members of own staff could be trying to undermine him.
"'YOU'RE FIRED' Any free agent leaker(s) on @realDonaldTrump's National Security Council w/their own agenda/vision for Presidency & America need to be tossed to curb in a NY minute," Rep. Lee Zeldin (R-N.Y.) tweeted Wednesday.
Zeldin suggested that Obama holdovers on the National Security Council may have been behind the breach. Trump and his allies have frequently warned that a "deep state" of government officials in Washington have constantly tried to torpedo his agenda.
"The 'what would @brhodes want me to do' attitude by some in NSC is a disease easily curable," the New York lawmaker tweeted, referring to Obama deputy national security adviser Ben Rhodes.
The person or people responsible for the leak have not been publicly identified.
The leak added to the wave of negative coverage of Trump's phone call with Putin, frustrating the president, who has faced Russia-related scrutiny ever since he was elected.
The Washington Post and other outlets reported Trump had been warned in briefing materials not to congratulate the Russian leader on his reelection. Aides even included the message "DO NOT CONGRATULATE" in the president's talking points.
But Trump went ahead and did it anyway, drawing criticism from Republicans on Capitol Hill.
A leading Putin opposition figure was barred from running in the election and international organizations reported instances of ballot-box stuffing and other irregularities during last Sunday's contest.
Trump also did not confront Putin over a nerve-agent attack against a former Russian double agent in Britain, despite being instructed in briefing papers to do so. Both the U.S. and U.K. have blamed the attack on Russia. Moscow has denied the accusation.
But Sen. Marco Rubio (R-Fla.), a critic of Trump's Russia stance, said Wednesday he was even more upset with the leaks than what the president did or did not say to the Russian leader.
"No, I don't like that he did it, but you know what I like even less? That there's someone close to him leaking this stuff out," Rubio told reporters at the Capitol.
"If you don't like the guy, quit. But to be this duplicitous and continue to leak things out, it's dangerous," the senator added.
The leak appeared to reveal deep frustration among some members of the administration over Trump's approach toward Russia.
Trump has long argued that forming a good relationship with Putin will help repair strained ties between the U.S. and Russia, despite warnings from advisers, lawmakers and allies that such an outcome is unlikely.
"The Fake News Media is crazed because they wanted me to excoriate him. They are wrong!" Trump tweeted Wednesday afternoon, referring to Putin. "Getting along with Russia (and others) is a good thing, not a bad thing," he continued. "They can help solve problems with North Korea, Syria, Ukraine, [Islamic State in Iraq and Syria] ISIS, Iran and even the coming Arms Race."Trump did not address the leak of his briefing materials on Twitter. He also did not appear in public Wednesday after his daily schedule was scrapped due to a snowstorm in Washington.
It's not clear whether the president had read or absorbed the warning from his staff about his call with Putin. The Washington Post reported that national security adviser H.R. McMaster briefed the president orally before the call, but did not raise the caution against a congratulations message.
McMaster, who is reportedly on the verge of being replaced, and others will likely be the subject of scrutiny in any probe of the leaks.
Attorney General Jeff Sessions announced the administration was cracking down on unauthorized disclosures last summer after the Post published full transcripts of Trump's private calls with the leaders of Mexico and Australia.
The Department of Justice (DOJ) on Wednesday did not comment on the effort. A spokesman said the agency "takes unauthorized leaks extremely seriously" and referred The Hill to Sessions's remarks from last summer.
At the time, Sessions said the DOJ had tripled the number of active leaks investigations and had created a counterintelligence unit to manage the heavy case load.
There have been few bombshell indictments since then, although one former National Security Agency contractor has been charged with the illegal "transmission of national defense information."
Comment: See also: CNN: Trump, Kelly furious over leak that Trump was not to congratulate Putin