One Fox executive said that the jabs by the White House could solidify the network’s audience base and recalled that Mr. Ailes had remarked internally: “Don’t pick a fight with people who like to fight.” The executive asked not to be named while discussing internal conversations.

Certainly, Fox continues to aggressively bolster its on-air talent, most recently with the hiring of John Stossel, the libertarian investigative journalist from ABC News, for its spin-off channel, Fox Business. The business channel is also keen on another administration critic, Lou Dobbs, who met for dinner with Mr. Ailes last month, according to two people with direct knowledge of the meeting.