WICHITA, Kan.—As Kansas Republicans brace themselves for a bitter primary battle over their state’s open U.S. Senate seat, the candidate generating the most buzz is the one who says he isn’t running.

U.S. Secretary of State Mike Pompeo told the Economic Club of Washington on July 29 that a Senate bid was “off the table.” But as Mr. Pompeo returned to his home state last weekend for a rare vacation, Kansans who have noticed that he keeps finding ways to nurture his local ties wondered whether he might still run—and if not,...