President Trump and his top advisers said Tuesday that he had a right to share counterterrorism information with Russian officials, as the White House considered efforts to root out anonymous leakers of classified intelligence who have undermined his presidency from the start.

Addressing reporters at the White House, Mr. Trump said his meeting last week with Russian Foreign Minister Sergey Lavrov will produce a more effective counterterrorism partnership with Moscow against the Islamic State group in the Middle East.

“We had a very, very successful meeting with the foreign minister of Russia,” Mr. Trump insisted. “Our fight is against ISIS. We’re going to have a lot of great success over the next coming years. And we want to get as many to help fight terrorism as possible.”

Referring earlier to a published report that accused Mr. Trump of providing highly classified intelligence to Russians, the president said on Twitter that he has “the absolute right” to share information on terrorism and airline safety that could help protect Americans.

“I want Russia to greatly step up their fight against ISIS & terrorism,” Mr. Trump said.

White House National Security Adviser H.R. McMaster, who attended the meeting with the Russians, said the president’s disclosures were “wholly appropriate” to advance U.S. national security.

“The president in no way compromised sources or methods,” Mr. McMaster said. “The president wasn’t even aware where this information came from. He wasn’t briefed on the source or method of the information, either. Information on this topic of the threat to aviation was shared with multiple allies.”

White House officials refused to say whether the counterterrorism intelligence, some of which was provided by Israel, was considered classified. But as the White House became sidetracked by the story for a second straight day, presidential aides and some U.S. intelligence officials said the episode was consistent with the pattern deep-state opposition by Obama-era holdovers in government who are trying to thwart Mr. Trump’s agenda by leaking unfavorable stories about him, especially about ties to Russia.

White House press secretary Sean Spicer called the latest leak to the media dangerous and said Mr. Trump intends to do everything he can to find the source. He refused to say whether the White House was conducting an active investigation into leaks.

“Somebody’s selectively leaking information and facts,” Mr. Spicer said. “It appears that somebody’s trying to create a narrative, or a problem. This is clearly a pattern of people releasing sensitive information to further what appears to be somebody’s agenda.”

A senior U.S. defense intelligence analyst agreed.

“The track record since Trump took office is the leakers have been out to make him look bad about any and everything he does,” said the official, who requested anonymity to protect his job. “There are certainly people in the intelligence community at a fairly senior level who are just trying to release anything that could be damaging to him. I don’t know how any sane person could not see that there’s some validity to the fact there are people out there who are hellbent on placing land mines in front of the Trump administration to prevent them from being successful.”

The analyst said the Obama administration was worse about leaking intelligence, especially regarding counterterrorism operations.

He said it has become common knowledge that counterterrorism officials use cellphone signals to locate terrorism suspects.

“Why do people know that? Because [Obama administration officials] couldn’t keep their damned mouths shut and talked to the press about how we use signals intelligence as a means to find people that we can then capture or kill,” he said. “The previous administration released tons of signals intelligence capabilities because it made them look good, when they killed or captured some bad guy.”

Victor Asal, a homeland security scholar at the Rockefeller College of Public Affairs and Policy at the State University of New York’s University at Albany, said he couldn’t decide which was worse: the accusations of Mr. Trump’s loose talk with Russians or the constant leaks to news media.

“They are both a mess,” he said.

Mr. Asal worried that shared information could hurt intelligence collection, but he said the leaks about discussions in the Oval Office were just as problematic.

“The fact that the White House seems to becoming a sieve leaking information that the president would like to be kept private is also highly problematic for management and image of the president and his team,” he said.

Rep. Steve King, Iowa Republican, said the leaks appeared to be coming out of the CIA, not the White House. He blamed Obama administration holdovers and other deep-state operators trying to undermine the Trump presidency.

“Trump has tried to work with some of the Obama holdovers. You will find, I suspect, the source of the leak is likely to be an Obama holdover or somebody who is just undermining this administration,” Mr. King said. “You can’t have that at the White House.”

He said he hoped Mr. Trump would begin rooting out his opponents within the intelligence community and start firing people before they fatally wound the presidency.

“They’ve bogged him down, but they are a long way from succeeding. From what I’ve seen of Donald Trump is, whenever things are down, he has a way of turning it around and making a silk purse out of a sow’s ear. I’m going to put my money on Trump, but I think he needs to shake things up,” Mr. King said.

Todd Weiler, a former assistant secretary of defense, said the fallout from accusations that Mr. Trump divulged intelligence gathering sources and methods would impact U.S. military and other relationships with allies for years to come.

“The damage is done,” said Mr. Weiler, who served in top Pentagon posts in the Obama and Clinton administrations. “A lot of what makes this harmful is the timing. There continues to be so much smoke [about Russia] that so many people think there has to be fire.”

He said the leaks from the intelligence community, if that’s the source, were likely acts of self-preservation.

“I’m not a believer in leaks, but in this city it happens a lot,” Mr. Weiler said.

He blamed Mr. Trump for being reckless and said that if Hillary Clinton were president and did the same thing, Republicans would be demanding impeachment and that she be locked up.

Mr. McMaster said leaks are “the real issue.”

“Our national security has been put at risk by those violating confidentiality, and those releasing information to the press that could be used, connected with other information available, to make American citizens and others more vulnerable,” he said.

On Capitol Hill, lawmakers in both parties expressed concern about the president’s actions, leaks and yet another chaotic week at the White House.

Sen. John McCain, Arizona Republican, called reports that Mr. Trump revealed intelligence “deeply disturbing” and said they could affect the willingness of U.S. allies to share intelligence with the U.S.

Senate Majority Leader Mitch McConnell, Kentucky Republican, said the episode was creating a distraction from priorities in Congress such as tax reform and replacing Obamacare.

“I think we could do with a little less drama from the White House on a lot of things so that we can focus on our agenda,” he told Bloomberg Business.

Senate Minority Leader Charles E. Schumer, New York Democrat, called for Congress to have immediate access to a transcript of Mr. Trump’s meeting with the Russians. The White House won’t say whether such a transcript exists.

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