If there was a wiretap order targeting Mr. Trump or his associates, what would that mean?

If it was a criminal wiretap, it would mean that the Justice Department had gathered sufficient evidence to convince a federal judge that someone using the phone number or email address probably committed a serious crime. If it was a national security wiretap, it would mean a federal judge on the Foreign Intelligence Surveillance Court had a basis to believe the target was probably an agent of a foreign power, like Russia.

Could a judge have approved tapping something at Trump Tower for another reason?

Yes. For example, FISA orders have two parts. After determining that there is probable cause to believe that the target is a foreign agent, a judge also has to approve directing surveillance at a particular “facility,” like a phone line or an email address, that the target is probably using to communicate. So in theory, if there was reason to believe that some other lawful target was communicating from Trump Tower, a judge could have authorized surveillance at a facility there for that reason.

What about the computer server registered to Trump Tower?

Several news media outlets have reported that investigators last year were puzzled by data transmissions between a computer server registered to Trump Tower and a computer server associated with a Russian bank. Although Mr. Trump on Twitter talked about his “phones,” in theory a judge might determine that the computer address of the server was a facility being used by a foreign power, Russia, to communicate, and authorize surveillance of it.

Isn’t there a report about an October surveillance court order involving that server?

Yes. The Breitbart story relied heavily on a Nov. 7 article by a British writer on the news and opinion website HeatStreet. It claimed that the Foreign Intelligence Surveillance Court had approved a “warrant” in October in connection with activity between Russian banks and the server, which it said — and Breitbart repeated — was in Trump Tower. It further stated that “it is thought in the intelligence community” that this purported warrant permitted the collection of emails and other communications of Americans connected to the server investigation, which “thus covers Donald Trump.”

As things stand, there are reasons to be skeptical. HeatStreet had vague sourcing — two “sources with links to the counterintelligence community” — and it does not regularly publish investigative stories about American intelligence or law enforcement operations. To date, reporters for The New York Times with demonstrated sources in that world have been unable to corroborate that the court issued any such order. (Computer specialists have also pointed out that the server in question does not appear to be located in Trump Tower.) On Sunday, James Clapper, who was the director of national intelligence until Jan. 20, denied to NBC News that the Foreign Intelligence Surveillance Court had approved monitoring at Trump Tower.