The first time they met was in Germany. President Donald Trump took his interpreter’s notes afterwards and ordered him not to disclose what he heard to anyone. Later that night, at a dinner, Mr Trump pulled up a seat next to Russian President Vladimir Putin to talk without any American witnesses at all.

Their third encounter was in Vietnam, when Mr Trump seemed to take Mr Putin’s word that he had not interfered in U.S. elections. A formal summit followed in Helsinki, where the two leaders kicked out everyone but the interpreters.

Most recently, they chatted in Buenos Aires, Argentina, after Mr Trump said they would not meet because of Russian aggression.

Mr Trump has adamantly insisted there was “no collusion” with Russia during his 2016 presidential campaign. But each of the five times he has met with Mr Putin since taking office, he has fueled suspicions about their relationship.

The unusually secretive way he has handled these meetings has left his own administration guessing what happened and piqued the interest of investigators.

“What’s disconcerting is the desire to hide information from your own team,” said Andrew S. Weiss, who was a Russia adviser to President Bill Clinton. “The fact that Mr Trump didn’t want the State Department or members of the White House team to know what he was talking with Mr Putin about suggests it was not about advancing our country’s national interest but something more problematic.”

The mystery surrounding the meetings seems to have drawn attention from the special counsel, Robert Mueller, who is examining ties between the president and Russia. And it has generated a furor in Congress, where Democrats are pushing to subpoena the notes of the president’s interpreters or perhaps the interpreters themselves.

Veterans of past administrations could not recall a precedent for a president meeting alone with an adversary and preventing his own advisers from being briefed on what was said. When they meet with foreign leaders, presidents typically want at least one aide in the room — not just an interpreter — to avoid misunderstandings later.

Donald Trump insists 'no collusion' after Michael Flynn admits lying to FBI

“All five of the presidents whom I worked for, Republicans and Democrats, wanted a word-for-word set of notes, if only to protect the integrity of the American side of the conversation against later manipulation by the Soviets or the Russians,” said Victoria J. Nuland, a career diplomat who worked for Dick Cheney and Hillary Clinton, among others.

That would seem an even greater imperative for Mr Trump, who knew there were questions about his relationship with Mr Putin given that U.S. intelligence agencies concluded that Moscow tried to help elect him.

“If any president would have wanted witnesses and protection, it ought to have been Donald Trump,” said Richard N. Haass, the president of the Council on Foreign Relations and adviser to four presidents, most recently as President George W. Bush’s State Department policy planning director. “And yet he chose not to, and that adds fuel to the fire that something here is not right.”

Mr Trump’s defenders acknowledge his approach does not resemble the way his predecessors operated, but note that he has been an unorthodox president in so many ways that it does not prove anything untoward. And, they say, he has reason to feel burned since previous interactions with foreign leaders have leaked, including full transcripts of telephone calls with the leaders of Mexico and Australia published in The Washington Post.

“Of course I was disappointed with Helsinki, but I do not just look at how the president handles specific meetings with Putin,” said Luke Coffey, a foreign policy scholar at the Heritage Foundation. “Instead, I’m most interested in what the actual policies are coming out of the administration.”

He cited additional sanctions, weapons sent to Ukraine, increased Pentagon spending meant to counter Russian aggression and opposition to a new Russian pipeline to Europe. All that, he said, “is proof that this is one of the toughest administrations on Russia since Reagan.”

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The question of Mr Trump’s meetings with Mr Putin was revived by a pair of news stories last weekend. The New York Times reported that after Mr Trump fired FBI Director James B. Comey in 2017, the bureau opened a counterintelligence investigation to explore whether the president was acting on Russia’s behalf. The Post reported that Mr Trump had gone to unusual lengths to conceal details of his talks with Mr Putin, including taking his interpreter’s notes.

The White House dismissed the stories as unfair smears. “The liberal media has wasted two years trying to manufacture a fake collusion scandal instead of reporting the fact that unlike President Obama, who let Russia and other foreign adversaries push America around, President Trump has actually been tough on Russia,” Sarah Huckabee Sanders, the White House press secretary, said in a statement.

Mr Trump has been in contact with Mr Putin since shortly after his election in November 2016. Mr Putin sent him a congratulatory telegram and the two spoke by telephone on 14 November.

They spoke a few more times before meeting in person for the first time as presidents on 7 July 2017, in Hamburg, Germany, during a Group of 20 economic summit. Aside from interpreters, the only others in the room were Rex W. Tillerson, then the secretary of state, and Sergey V. Lavrov, the Russian foreign minister.

The inaugural meeting came at a sensitive time. Mr Trump’s team learned that day that one of the biggest secrets of his presidential bid was about to become public: At the height of the campaign, his son, son-in-law and campaign chairman had met at Mr Trump Tower with Russians on the promise of obtaining dirt on Hillary Clinton from the Russian government. Mr Trump’s team was scrambling to respond to a request for comment by The Times.

Mr Trump’s meeting with Mr Putin that day lasted more than two hours. Afterward, Mr Trump took his interpreter’s notes and instructed the interpreter not to brief anyone. Mr Tillerson told reporters that the leaders discussed everything from Syria to Ukraine, but he also described “a very robust and lengthy exchange” on the election hacking.

A few hours later, Mr Trump sought out Mr Putin again during a dinner for all the leaders. Videotape later made public showed Mr Trump pointing at Mr Putin, who was seated across and down a long table, then pointing at himself and then making a pumping motion with his fist.

Mr Trump later told The Times that he went over to see his wife, Melania Trump, who was sitting next to Mr Putin, and the two leaders then talked, with Mr Putin’s interpreter translating. No US officials were present, and the White House did not confirm the encounter until more than 10 days later, after it was independently reported.

The day after the two meetings, as Mr Trump was on Air Force One taking off from Germany heading back to Washington, he telephoned a Times reporter and argued that the Russians were falsely accused of election interference. While he insisted most of the conversation be off the record, he later repeated a few things in public in little-noticed asides.

He said that he raised the election hacking three times and that Mr Putin denied involvement. But he said Mr Putin also told him that “if we did, we wouldn’t have gotten caught, because we’re professionals.” Mr Trump said: “I thought that was a good point because they are some of the best in the world” at hacking.

Asked how he weighed Mr Putin’s denials against the evidence that had been presented to him by Mr Comey; John O. Brennan, then the CIA director; and James R. Clapper Jr., then director of national intelligence, he said that Mr Clapper and Mr Brennan were the “most political” intelligence chiefs he knew and that Mr Comey was “a leaker.”

Later on the same flight to Washington, Mr Trump huddled with aides to decide how to respond to the emerging story by other Times reporters about the Trump Tower meeting. He personally dictated a misleading statement, saying the meeting was about Russian adoptions without admitting that it was actually intended to accept Moscow’s aid for his campaign, as emails obtained by The Times later documented.

The confluence of the two conversations with Mr Putin even as Mr Trump’s team was grappling with questions about the Trump Tower meeting have fueled further suspicions.

“If you add up all these pieces, it’s a very damning picture at a minimum of how to handle national security,” said Weiss, who is now at the Carnegie Endowment for International Peace. “If there’s a more nefarious explanation, it’s obviously more disturbing.”

Mr Trump next encountered Mr Putin in person on 11 November 2017, at a meeting of the Asia-Pacific Economic Cooperation forum in Da Nang, Vietnam. No formal meeting was scheduled, but the two chatted anyway, and Mr Trump later indicated that Mr Putin again denied any election interference. “I really believe that when he tells me that, he means it,” Mr Trump said.

The two stayed in touch by phone. Mr Trump called after Mr Putin was re-elected in a contest heavily managed by the state in his favor. Although wary aides wrote in his briefing papers, “DO NOT CONGRATULATE,” Mr Trump went ahead and congratulated Mr Putin anyway.

US President Donald Trump meets with Russian President Vladimir Putin in Helsinki, Finland, 16 July 16, 2018. (file photo:Kevin Lamarque/REUTERS) ((file photo:Kevin Lamarque/REUTERS))

Their most famous meeting came 16 July 2018, in Helsinki, where they talked for more than two hours accompanied only by interpreters. At a subsequent news conference, Mr Trump seemed to again accept Mr Putin’s denial of election interference over the conclusions of American intelligence agencies.

But what happened behind closed doors remained shrouded. The Kremlin later reported that the leaders reached important agreements, but U.S. government officials were left in the dark. U.S. intelligence agencies were left to glean details about the meeting from surveillance of Russians who talked about it afterward.

Within months, Mr Trump was angling for another meeting, perhaps at the White House or in Paris. Finally, they scheduled a get-together in Buenos Aires in December on the sidelines of yet another G-20 meeting.

President Donald Trump and Russia's President Vladimir Putin as they gather for a group photo at the start of the G20 summit, Buenos Aires, Argentina. 30 November 2018. (AP Photo/Pablo Martinez Monsivais, File) ((AP Photo/Pablo Martinez Monsivais, File))

Days before, Russian forces seized three Ukrainian naval vessels, but Mr Trump seemed intent on sitting down with Mr Putin anyway, telling reporters as he left the White House for Buenos Aires that the meeting was still on. Just an hour later, after aides briefed him again on the Ukraine standoff, he canceled the meeting on Twitter, catching the Russians off-guard.

But when he arrived in Buenos Aires, Mr Trump ended up having another informal conversation with Mr Putin at the leaders’ dinner. Once again, little information emerged about what they discussed, even to other U.S. government officials.

“I’ve never heard of a president conducting one-on-one meetings with his Russian counterpart without note-takers or without afterward offering readouts to his top aides,” said David J. Kramer, a former assistant secretary of state under Bush. “Putin is privy to what the two discussed — why can’t senior administration officials be trusted and looped in, too?”