“Things are trending blue,” said Barry Grissom, one of the Democrats who is running and who described the timing as a “perfect opportunity.” He predicted Mr. Pompeo would ultimately not enter the race.

“He may be more versed on what’s transpiring in Afghanistan and Iraq,” said Mr. Grissom, a former United States attorney during the Obama administration. “But I question if he knows what’s going on in Dodge City or Garden City — issues that really concern Kansas families.”

State Senator Barbara Bollier, an anesthesiologist who left the Republican Party in December to become a Democrat, is also considering running and has been wooed by Senator Chuck Schumer of New York, the minority leader, and other high-profile Democrats. If she does run, she most likely would be backed by national Democratic leaders and Emily’s List, the political action committee that donates to Democratic women who support abortion rights.

“I don’t know Mike Pompeo personally,” Ms. Bollier said. “We all see he is very strongly affiliated with President Trump, and I would guess those Kansans who want more Trump politics would like him as a candidate.”

Both Mr. Grissom and Ms. Bollier are from moderate Johnson County in the Kansas City suburbs — where Ms. Kelly won enough votes to defeat Mr. Kobach last year. But they face their own challenges: Ms. Bollier remains undecided about running in a federal election, and Mr. Grissom has been questioned over his management of the United States attorney’s office after prosecutors were accused of violating defendants’ rights.

Not one of the Democrats has the statewide name recognition that Mr. Pompeo commands.

Mr. McConnell frequently calls Mr. Pompeo to make his pitch, and has dispatched others in the party, like Senator Tom Cotton of Arkansas, to do so, as well, according to senior Republican officials. They described a June poll by the National Republican Senatorial Committee that showed Mr. Pompeo handily defeating all other Republicans in a primary.

“I see that as a real potential, that the secretary may decide that he would be interested in serving in the United States Senate, and I think he would be a solid candidate,” Senator Jerry Moran, the junior Republican senator from Kansas, told reporters in Wichita on Thursday.