Mr. Jones said that conversation had taken place earlier in the campaign, not on the phone call immediately after the election that my colleague Maggie Haberman reported on, in which Mr. Jones said the president had thanked him for his support. Mr. Jones told me that he had spoken with Mr. Trump since that call, though an aide to the president, communicating on the condition of anonymity, played down the frequency of their contact.

Either way, Mr. Jones is hoping his organization will qualify for a coveted White House press credential. He says it’s not something he’s pining for or needs, but he doesn’t see why Infowars shouldn’t get one when “Trump’s calling CNN fake.”

The White House said it had yet to receive a proper application from Infowars and therefore could not comment on whether it would get one. Mr. Jones said the delay might be related to a bureaucratic snag. “They say it’s going to get rectified,” he said.

One ally in his corner is Roger Stone, the longtime Republican operative and informal adviser to Mr. Trump, whose matchmaking brought them together and led to the 2015 Infowars interview during which Mr. Trump told Mr. Jones that “you have an amazing reputation.”

Mr. Stone said in an interview, “They are reaching millions of people, and these people are steadfast and loyal Trump supporters.”

Two of the major internet tracking companies, Quantcast and Alexa, reported that in January Infowars had an average of around eight million (Quantcast) or 8.7 million (Alexa) global visitors, who viewed its pages nearly 50 million times. As of Sunday Quantcast ranked its traffic above that of the fact-checking site Politifact.com.

Those numbers miss the audiences for his national radio show and his team’s videos on YouTube, where the biggest of his 18 channels has 1.2 billion views, and on Facebook, where they draw many millions of views. (One, by his editor at large, Paul Joseph Watson, lists 18.1 million views.)