WASHINGTON — The first time they met was in Germany. President Trump took his interpreter’s notes afterward 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 President Vladimir V. 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 American elections. A formal summit meeting followed in Helsinki, Finland, where the two leaders kicked out everyone but the interpreters. Most recently, they chatted in Buenos Aires 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 many in 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 Trump didn’t want the State Department or members of the White House team to know what he was talking with Putin about suggests it was not about advancing our country’s national interest but something more problematic.”