The search for Jon Stewart’s successor appears to be intensifying.

With several obvious candidates for Stewart’s “Daily Show” seat ruling themselves out as candidates, the likelihood is growing that Comedy Central will have to give stronger consideration to an external applicant.

One contributor to “The Daily Show,” Jessica Williams, recently took to Twitter to declare herself “unqualified” for the job. Jason Jones, one of the program’s faux correspondents, recently announced he would leave the program to star in and produce a new series for Time Warner’s TBS. His wife, longtime “Daily Show” player Samantha Bee, will stay with the program but will also have a producer role on the new show.

Other people who have been cited as potential successors in social-media speculation are also out of the running: Joel McHale is currently under contract at E!, and John Oliver, a former “Daily Show” contributor, recently extended his stay with HBO through 2017.

The looming choice is crucial for executives at the Viacom-owned network. Stewart is viewed as a Walter Cronkite of sorts by a rising generation, riffing on news-making subjects of the day and tilting at the way mainstream news outlets like CNN and Fox News deliver information. His program has been a linchpin of the network’s schedule. What’s more, Comedy Central recently bid farewell to Stephen Colbert, host of “Daily Show” companion “The Colbert Report,” and replaced it with “The Nightly Show,” a program hosted by Larry Wilmore that relies more heavily on discussion of topics relating to culture, race and gender.

The network is believed to have a “short list” of candidates and is moving toward making a decision, though it is not under pressure to do so without being methodical. Stewart has said he is likely to stay in “The Daily Show” chair until at least sometime in July and may even remain at the desk through the end of the year.

The show’s history would suggest Comedy Central is seeking someone in their 30s, just as both Stewart and his predecessor, Craig Kilborn, were when they took up the program’s reins. Under Kent Alterman, president of original programming, Comedy Central has in recent years sought out people who have a unique voice or world view, including Amy Schumer or Abbi Jacobson and Ilana Glazer of “Broad City.” For Stewart’s eventual successor, a distinct take on culture, news and politics may be more important than relative fame.