If you like New Zealand enough to want to become a citizen, the country’s Internal Affairs Department noted on Wednesday, one requirement is “to have been physically in New Zealand for a minimum of 1,350 days in the five years preceding the citizenship application.” Another requirement is that you “continue to reside” there after becoming a citizen.

Mr. Thiel, 49, does not appear to have done either.

The investor, who retains his American citizenship, was a founder of the online payments site PayPal and the data company Palantir. He secretly funded the lawsuit that killed off Gawker Media, the network of gossip sites that outed him as gay.

When Mr. Trump won, Mr. Thiel emerged as a key adviser. He has spent much of the time since the election in New York as part of the transition team. People from Mr. Thiel’s network are under consideration for significant jobs in Trump’s cabinet.

As a byproduct of his singular support for Mr. Trump in Silicon Valley, Mr. Thiel has become famous, a fate many of his peers go out of their way to avoid. He has been reported as a possible Supreme Court justice, as a potential Republican candidate for governor of California, and most recently, as President Trump’s potential ambassador to Germany. (He denied the first, and the others appear unlikely.)

Mr. Thiel’s admiration for New Zealand is longstanding. “Utopia,” he once called it. He has an investment firm in the country that has put millions into local start-ups. He also owns lavish properties there, which his Silicon Valley friends hope to fly to in the event of a worldwide pandemic.