Last month, Paolo Ardoino, the chief executive of the cryptocurrency company Tether, joined business executives and U.S. lawmakers for a private lunch at the Willard, a luxury hotel near the White House.

For years, Tether had faced accusations that it lied about its finances and allowed crime to flourish on its platform. But at the Willard, Mr. Ardoino and other crypto leaders were warmly greeted by Senator Bill Hagerty, a Tennessee Republican on the Senate Banking Committee, who attended the lunch and participated in a group discussion about digital currency regulations and national security, according to four people with knowledge of the event.

The episode was a sign of the changing landscape for crypto firms as President Trump has embraced the industry. Once an elusive overseas operator with little public footprint in the United States, Tether is capitalizing on the shift to establish a presence in Washington.

Since Mr. Trump’s inauguration, Tether has

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