WICHITA — Secretary of State Mike Pompeo toured a workforce development program in his adopted hometown Thursday, refueling speculation amid an expanding impeachment inquiry of President Trump that he may leave the administration to run for the Senate next year.

Pompeo — who launched his political career in Wichita, riding the tea party wave to Congress in 2010 — observed student welders and held roundtable discussions with students at WSU Tech with Trump’s daughter Ivanka. The event had the air of a school board campaign stop rather than a highflying diplomatic conclave, with Pompeo posing for photographs and looking on as Ivanka Trump, donning safety glasses, tried her hand at riveting.

Earlier in the day, a Wichita Eagle reporter asked if touring a technical school was the best use of time for America’s chief diplomat. Pompeo said “absolutely, 100 percent,” adding that he also wanted to attend the wedding of his son’s best friend.

Pompeo declined to answer questions about the impeachment inquiry, and when asked whether the treatment of Kurdish allies in Syria had “undercut U.S. credibility,” he shot back — “the whole predicate of your question is insane.”

Pompeo, 55, came home to a state that will probably remain resolutely red in next year’s presidential election, where locals are expressing “a lot of frustration” with the Democratic impeachment inquiry in Washington, said Rep. Ron Estes (R-Kan.), who appeared with Pompeo at the event.

“There are just so many other things we need to do,” Estes said.

Despite Pompeo’s repeated attempts to tamp down speculation on a possible Senate bid — “it’s off the table” he said in August — he has sometimes couched his denials in vague enough terms to suggest he is trying to keep his options open.

He said Thursday that there had been “no change” in his thinking. His “mission” is “to ­execute American diplomacy, to make sure that American markets are open for Kansas products all around the world,” he said. “That’s what I’m focused on. And it’s what I continue, intend to continue to be focused on.”

It was his fourth trip to the state this year, and “the timing is eyebrow-raising,” said Brandon Woodard, a Democratic state legislator from the Kansas City area.

Pompeo, who previously served as CIA director under Trump, arrived at the State Department last spring promising to restore “swagger” to beleaguered diplomats, who had felt shut out of policymaking under his predecessor, Rex Tillerson.

But as House committees conduct an impeachment inquiry of Trump, testimony from a succession of State Department witnesses has laid bare the growing level of discontent with Pompeo’s stewardship of the agency.

Testimony by a number of State Department officials and ­ former employees has raised even more questions about how much Pompeo knew about the Trump administration’s efforts to pressure Ukrainian officials to investigate his chief political rival, former vice president Joe Biden. Pompeo has said he will not speak publicly about private conversations.

“He’s done everything possible to keep his head down during this Ukraine stuff,” said a former diplomat, speaking on the condition of anonymity to be candid. “He almost certainly will run for the Senate in Kansas. So he wants to be sure he has the complete and vocal backing of the president. This Ukraine thing, this ­God-awful mess, could cause him a problem.”

Around the State Department, it is widely believed that Pompeo is still mulling his candidacy. But it is uncertain what impact the House impeachment inquiry will have on his chances among Kansas voters.

The impeachment inquiry “hasn’t fundamentally changed anything on the ground in regard to Pompeo,” said Russell Arben Fox, a political science professor at Friends University in Wichita. “He’s the sort of Republican Republicans really like. He has this reputation for being smart and stable and conservative — but not weird. And he could straddle the fence — appease both moderate and more conservative factions of the Republican Party.”

Republican leadership — including Senate Majority Leader Mitch McConnell (R-Ky.) — has been urging Pompeo to consider running since Sen. Pat Roberts (R-Kan.) announced earlier this year that he would be retiring.

“The field is just very unsettled, and Kansas Republicans are anxious for a clear conservative leader — Secretary Pompeo is where everyone wants to be,” said Jared Suhn, a political consultant who previously worked for Roberts.

Though Kansas has a late ­primary-filing deadline of June 1, Pompeo could not realistically postpone a decision until then and keep other Republican potential candidates waiting for him to make up his mind, analysts said. The establishment front-runner, Rep. Roger Marshall, a Republican in western Kansas, already has nearly $2 million on hand.

Republican leaders are particularly concerned about the candidacy of Kris Kobach, the former Kansas secretary of state who lost the governor’s race to Democrat Laura Kelly last year. Kobach’s ­anti-immigrant views and penchant for Trumpian grandstanding could turn off some moderate voters if he prevailed in the primary contest, analysts say.

Kobach “has been a polarizing figure,” said Patrick Miller, a political science professor at the University of Kansas. “They’re afraid he could put a Senate seat in play when the seat should be safely Republican.”

Rumors about Pompeo’s ambitions have been stoked in part by a string of interviews he has given to Kansas media and his frequent trips home to the state, where he spent over a decade as a businessman and forged ties to conservative billionaire industrialists Charles and David Koch. He has maintained his Kansas connections in other ways, such as appearing at the annual dinner of the Kansas Society of Washington this summer and meeting with the Kansas contingent at a Veterans of Foreign Wars conference in Orlando.

Analysts say if he doesn’t run next year he may eye a Senate race in 2022 or even presidential bid in the future, with a résumé that also includes the U.S. Military Academy, a stint in the Army and Harvard Law School.

Pompeo’s leadership at the State Department has come under fire since diplomats have testified about their concern over efforts by Trump and his personal lawyer, Rudolph W. Giuliani, to get Ukrainians to investigate Biden and his son Hunter. William B. Taylor Jr., whom Pompeo pulled back from retirement to become the head of the U.S. Embassy in Kyiv, said he wrote a cable to Pompeo saying the efforts at “unofficial” diplomacy were “folly.”

This week alone, at least four groups of former diplomats from the State Department and the U.S. Agency for International Development have written open letters and issued statements expressing their disappointment that Pompeo has not defended diplomats from being mistreated. Marie Yovanovitch, who was abruptly recalled as the U.S. ambassador in Kyiv, has become a hero of the diplomatic corps.

“We have been inspired by the leadership these diplomats have shown, despite President Trump’s and Secretary Pompeo’s efforts to silence them,” Foreign Policy for America, a group led by former diplomats, said a statement issued Wednesday.

Many current employees say they share the sentiments but could not sign their names for fear of being fired. Morale has plummeted as many employees say they believe Pompeo has failed to publicly defend the diplomats who have testified despite his order not to and Trump’s dismissal of them as “radical” bureaucrats of the “deep state.”

Correction: An earlier version of this story misstated the timing of a possible Senate bid by Secretary of State Mike Pompeo. Analysts say he could run next year or in 2022.