Alex Trebek, the host of “Jeopardy!” since 1984, stoked speculation on Monday that he could retire when his contract expires in 2020, telling a TMZ host that his chances of staying longer were “50-50, and a little less.”

The absence of Mr. Trebek, 78, would fundamentally alter the lightning-fast trivia show, which relies on his steady delivery to keep the pace humming. His comments, made on a Fox News program hosted by Harvey Levin of TMZ, were far from conclusive, and fans have long guessed who his replacement might be only to see him renew his contract again. But this time he suggested two people he saw fit to replace him — neither of whom had been on the radar of most fans.

He first mentioned Alex Faust, the play-by-play announcer for the N.H.L.’s Los Angeles Kings. On Twitter, Mr. Faust appeared to be surprised but grateful for the plug.