WASHINGTON—President Donald Trump and his administration worked to contain the fallout Tuesday after reports that he disclosed sensitive counterintelligence to Russian officials, with the president himself tweeting that he has the “absolute right” to share such information.

The information that was shared was provided by Israel, according to officials with direct knowledge of the matter.

In a news briefing Tuesday, Mr. Trump’s national-security adviser, Lt. Gen H.R. McMaster, said Mr. Trump’s conversation “was wholly appropriate” but that he believed the leaking of it put national security at risk.

Gen. McMaster wouldn’t discuss whether information Mr. Trump conveyed to the Russian foreign minister and ambassador was classified, but said that the president had “in no way compromised any sources or methods in this conversation.” He said Mr. Trump hadn’t been briefed on the source of the intelligence he discussed.

Earlier Tuesday, Mr. Trump tweeted that he has the “absolute right” as president to share “facts pertaining to terrorism and airline flight safety,” before offering an explanation for why: “Humanitarian reasons, plus I want Russia to greatly step up their fight against ISIS & terrorism.”

Later Tuesday, after delivering joint remarks with Turkish President Recep Tayyip Erdogan at the White House, Mr. Trump briefly addressed his meeting with Russian officials last week, saying it had been “very, very successful.”

“We’re going to have a lot of great success,” Mr. Trump said. “We want to get as many to help fight terrorism as possible.” He then exited the event.

The escalating controversy drew sharp comments on Capitol Hill, where Democrats were already galvanized by Mr. Trump’s firing last week of James Comey as head of the Federal Bureau of Investigation, which is looking into alleged ties between Mr. Trump’s associates and Russia.

Rep. Adam Schiff (D., Calif.), the ranking Democrat on the House Intelligence Committee, who said he viewed the White House response to reports about Mr. Trump’s meeting with Russian officials as “non-denial denials.”

Central Intelligence Agency director Mike Pompeo is expected to brief members of the House committee on Tuesday, according to two people familiar with the matter.

Senate Minority Leader Chuck Schumer (D., N.Y.) called on Mr. Trump to release the transcript of his meeting with Russian officials last week to congressional intelligence committees.

Some Republicans also expressed alarm, with Sen. John McCain (R., Ariz.) saying Tuesday that reports the president had shared sensitive intelligence from a U.S. ally with Russian officials were “deeply disturbing” and send “a troubling signal.”

Sen. Richard Burr (R., N.C.), chairman of the Senate Intelligence Committee, said his panel hadn’t yet been in contact with the White House but hoped to reach them by the end of Tuesday. He also said that he was unsure whether a transcript existed of the meeting with the Russians but said he was aware of notes on the Oval Office meeting.

The Wall Street Journal reported late Monday that Mr. Trump divulged details about Islamic State to the Russian foreign minister and ambassador in his meeting last week in a way that revealed enough information for them to potentially compromise the source of the intelligence, according to officials.

The Washington Post first reported that Mr. Trump had shared sensitive information with the Russians and said White House officials called the Central Intelligence Agency and National Security Agency to warn of Mr. Trump’s disclosure and its possible consequences.

Mr. Trump’s Tuesday tweets were a departure from a hastily assembled but circumspect White House response Monday night, when officials including Gen. McMaster said the Post article “as reported is false.”

Asked to specify Tuesday what was false, Gen. McMaster said that he disputed the article’s premise that Mr. Trump had acted inappropriately.

Gen. McMaster also said he was “not concerned at all” about the potential for U.S. information-sharing relationships to be in jeopardy and added that Mr. Trump “wasn’t even aware of where this information came from. He wasn’t briefed on the source.”

According to one U.S. official, the information shared was highly sensitive and difficult to acquire and was considered extraordinarily valuable. The Journal agreed not to identify the ally in its articles Monday. The Journal decided to name Israel once the country’s name was widely disseminated after reports by the New York Times and others Tuesday afternoon.

Gen. McMaster said Tuesday that Mr. Trump didn’t discuss Islamic State-controlled territories that weren’t already publicly known in his meeting with Russian officials last week.

“It was nothing that you would not know from open-source reporting,” Gen. McMaster said Tuesday of the president’s discussion of threats from Islamic State-controlled territories in his meeting with Russian Foreign Minister Sergei Lavrov and Russian Ambassador Sergei Kislyak. “It had all to do with operations that are already ongoing have been made public for months.”

Asked why White House homeland security adviser Tom Bossert had placed calls to the directors of the CIA and NSA after the meeting if Mr. Trump hadn’t disclosed any sensitive intelligence, Gen. McMaster said he hadn’t spoken with Mr. Bossert but that the adviser could have done so “out of an overabundance of caution.”

The White House didn’t provide a detailed statement about Mr. Trump’s meeting last week with Messrs. Lavrov and Kislyak, the Russian ambassador, which was closed to the press. A photographer from the Russian news agency TASS was in the room and published photographs.

Mr. Trump noted on Twitter that the meeting was “openly scheduled.” The meeting with Mr. Lavrov was on the president’s public schedule, but the schedule didn’t state that Mr. Kislyak would attend as well.

Later Tuesday morning, Mr. Trump wrote on Twitter that he had asked former FBI Director Comey and others “to find the LEAKERS in the intelligence community.....”

Presidents have the legal right to declassify intelligence as they see fit. But doing so can put intelligence sources abroad in danger and make them less willing to work with the U.S., several defense officials said, and the latest disclosures stunned Washington’s national-security veterans on both sides of the political divide.

South Dakota Republican Sen. John Thune called it “concerning” that “information that reveals classified national security information is shared with the Russians,” although he added that there was “conflicting information.”

Sen. Amy Klobuchar (D., Minn.) said the Trump administration’s initial pushback on the reports indicates the White House has a recording of the meeting.

“Clearly if there is some kind of a readout or a transcript from that meeting that means there is a tape,” Ms. Klobuchar said. “We need to get a hold of that” in the Senate.

Senate Majority Leader Mitch McConnell, for his part, said in a Bloomberg interview that he hoped to see fewer distractions coming out of the White House.

Donald J. Trump 

✔@realDonaldTrump

As President I wanted to share with Russia (at an openly scheduled W.H. meeting) which I have the absolute right to do, facts pertaining....

 Donald J. Trump 

✔@realDonaldTrump

...to terrorism and airline flight safety. Humanitarian reasons, plus I want Russia to greatly step up their fight against ISIS & terrorism.

Donald J. Trump 

✔@realDonaldTrump

I have been asking Director Comey & others, from the beginning of my administration, to find the LEAKERS in the intelligence community.....

9:10 AM - 16 May 2017

**Richard Rubin and Kristina Peterson contributed to this article.