“O’Reilly needs to have a platform and a voice — he’s got an important one — and we’re very open to having that discussion with him,” said Christopher Ruddy, the founder of Newsmax, a conservative-leaning news network and website. Mr. Ruddy, a close associate of President Trump, said his interest was “serious, and not hot air.”

Charles Herring, whose family company Herring Networks owns the newer and still smaller — but no less ambitious — conservative cable network One America News Network, said in an email, “One America News has reached out and spoken directly with Mr. O’Reilly and separately with his agent.”

And then there’s Sinclair Broadcast Group, which is already the country’s largest television station group but aims to become even bigger with its proposed takeover of Tribune Media. Sinclair, which has drawn scrutiny for pushing conservative-skewing segments onto its local newscasts, says it is not moving on Mr. O’Reilly. But no one is counting it out, including Rupert Murdoch, the 21st Century Fox founding chairman, who is paying close attention to Sinclair and Newsmax, I’m told by people with direct knowledge of his thinking.