Yet Ms. Goitein, who has no sympathy for Mr. Trump’s policies, believes his clumsy comments on wiretapping, even if not true, should be an opening for a broader discussion of government surveillance and American privacy. She is among the civil libertarians who believe Mr. Trump’s critics have been too quick to dismiss the real possibility that the National Security Agency or F.B.I. might actually have picked up Trump campaign communications under eavesdropping rules that civil libertarians see as too permissive.

“I don’t think we can laugh it off,” she said.

Intentionally or otherwise, Mr. Trump has rejuvenated the debate over the proper balance of privacy and security that surfaced after Edward Snowden’s disclosures about N.S.A. programs in 2013. When the libertarian Senator Rand Paul, Republican of Kentucky, used the Trump claims to suggest a broader concern about privacy, Glenn Greenwald, a left-wing writer for the online publication The Intercept, backed him up in a column titled “Rand Paul Is Right.”

“Paul’s explanation is absolutely correct,” Mr. Greenwald wrote. He said that the National Security Agency “is empowered to spy on Americans’ communications without a warrant,” calling current procedures a violation of the Fourth Amendment and “the dirty little secret of the U.S. Surveillance State.”

What these odd political bedfellows were pointing out is a truism inside the intelligence world but less understood outside it: When the National Security Agency or the F.B.I. eavesdrop on foreigners’ communications, they often pick up the Americans who are talking to them. National Security Agency and F.B.I. officials call this “incidental” collection, but it can have serious consequences.

It appears that such incidental collection, for instance, decided the fate of Michael T. Flynn, who stepped down as national security adviser after he was picked up talking to the Russian ambassador, Sergey I. Kislyak, and lied about the conversations.

To address the threat to American privacy from incidental collection, the government applies what it calls “minimization” rules. Names of Americans must be masked in intelligence reports disseminated by the agencies, but there are exceptions: Officials can request that the names be unmasked to help understand the reports, and the names are available to criminal investigators.

There is also the possibility of what is called “reverse targeting” — say, eavesdropping on Mr. Kislyak, ostensibly to find out what the Russian ambassador is up to — but with the real goal of catching Mr. Flynn. Reverse targeting is prohibited by law, but Ms. Goitein points out that it is difficult to prove because it requires showing what was in the eavesdropper’s mind.