Mayor Daniel Lurie of San Francisco did what he does best after the Trump administration prepared to send border agents to the Bay Area for a major crackdown on immigration this week.

He stayed calm. He worked the phones.

In doing so, Mr. Lurie and his team visualized concentric circles of prominent business leaders who could persuade President Trump and determined who might lobby the executives to make those important calls. They wanted to deliver the message, in the most diplomatic way possible, that San Francisco was not the apocalyptic landscape that the president sees on Fox News.

Soon, a host of billionaires with strong ties to the city were on the phone with Mr. Trump and Vice President JD Vance, a former venture capitalist in San Francisco.

Four people familiar with the situation described the City Hall discussions to The New York Times on condition of anonymity because they were not authorized to speak publicly.

Mr. Lurie, 48, a philanthropist and an heir to the Levi Strauss denim fortune, had never served in elected office until he became San Francisco’s mayor in January. To some fellow Democrats, he has been too scripted at a time that has called for brash leadership to counter Mr. Trump. Some have also resented his connections with tech billionaires.

But Mr. Lurie’s supporters say that his approach was simply an extension of his quiet leadership and his ability to work the levers behind the scenes in ways that few others can.

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