News of Mr. Carlson’s breakup with Fox immediately stoked speculation in right-wing media circles about his next steps. Mr. Carlson’s nightly perch on Fox News has made him a power broker and an opinion-shaper in the Republican Party. That would make him a serious get for the conservative media companies that have pitched themselves as alternatives to the mainstream press — and to Fox News, the right-wing media behemoth.

One associate of Mr. Carlson’s said that Rumble, a right-wing media company based in Canada, would be a likely suitor. The Daily Wire, a media start-up co-founded by the conservative commenter Ben Shapiro, is interested in exploring a deal with Mr. Carlson, as are the conservative TV networks Newsmax and One America News, according to three people familiar with their interest.

Fox News, for its part, is rotating hosts in Mr. Carlson’s old time-slot under the “Fox News Tonight” banner for the foreseeable future. So far, hosts have included Brian Kilmeade, who usually co-hosts the network’s popular “Fox and Friends” morning show, and Lawrence Jones, another host at the network.

Fox News, the longtime king of cable news ratings, has suffered an audience dip in Mr. Carlson’s hour after his abrupt departure. Fox’s audience in that time slot on some nights fell behind both CNN’s and MSNBC’s over the last week among adults ages 25-54, a rare loss in that demographic, which is coveted by advertisers.

For Mr. Carlson, there’s more at stake than a paycheck, Mr. Lee said. Many TV personalities become less relevant the longer they stay off the air, hurting their earning potential.

“It’s not always just a question of money, because being off the air for an extended period can have an adverse effect on a career,” Mr. Lee said.

Jim Rutenberg contributed reporting.