(CNN) —  

National Security Adviser Robert O’Brien said he’s spoken in the past few days with Secretary of State Mike Pompeo, who told him he’s not running for a US Senate seat in Kansas.

“I hope and I expect Secretary Pompeo will stay. I spoke with him about this two nights ago. He said he’s not running for Senate. He said he’s staying as secretary of state,” O’Brien said Sunday on ABC’s “This Week.”

Speculation has mounted that Pompeo may be preparing to leave his post to mount a run for Senate, fueled partly by the creation of new social media accounts casting the hard-nosed diplomat in a softer light.

Some Trump administration officials have been tossing around names for potential replacements, sources told CNN last week. They included O’Brien, as well as Treasury Secretary Steven Mnuchin and Deputy Secretary of State Stephen Biegun.

O’Brien said Sunday he “loves” the job he currently holds. “I’m happy sitting where I am,” he said.

President Donald Trump, who has said Pompeo would run if there were a risk of Republicans losing the Kansas seat, has received briefings from Senate Majority Leader Mitch McConnell on Pompeo’s polling strength in the state, people familiar with the matter have said.

Publicly, Pompeo has downplayed and even dismissed the idea that he is considering a run, despite taking regular trips to Kansas and sitting for interviews with Kansas media outlets.

On Monday, he again downplayed the prospect of a run, but did not dismiss it entirely.

“I’ve watched my life take turns that one would never have expected, but it’s not something I want to do. I want to stay here and continue to perform the mission that I’m serving President Trump, and I hope doing a good turn for the American people as well,” Pompeo said in an interview with Fox & Friends. Calling Kansas “home,” he said it’s his “intention to stay here and continue to serve as President Trump’s Secretary of State.”

“I’ve said that consistently,” Pompeo said in the interview. “I intend to keep saying it, and as long as President Trump wants me to serve in this capacity, there’s still work to do.”

But the new social media accounts aren’t helping to quiet the speculation. In his new online image, Pompeo is a homegrown, all-American family man – with almost no mention of his role in Washington. In addition to the photos of card games at his kitchen table and drinking beer, Pompeo is seen washing dishes in a pair of gym shorts and attending college football games.

Pompeo’s prospective run may have no bigger supporter than McConnell. The Kentucky Republican has been effusive in his praise for the former House member and unabashed about his desire to see Pompeo as the next senator from Kansas.

Repeatedly, McConnell has said publicly that he wants Pompeo to run. He invited Pompeo to speak at his alma mater, the University of Louisville, as part of a speaker’s series at the endowed institution that bears his name: the McConnell Center.

Kansas hasn’t had a Democratic senator since 1932, but Sen. Pat Roberts’ looming retirement has become a cause for concern within the GOP establishment.

Their worry is that the party will nominate Kris Kobach, an anti-illegal-immigration firebrand and voter-fraud crusader who lost the governor’s race last year to a Democrat. Kobach then turned around and announced his candidacy for Senate in July.

Democrats also made gains in the Kansas 2018 congressional races. The GOP worry is that a Senate race with Kobach could become a repeat of the governor’s race, paving the path for a Democratic victory.

CNN’s Kylie Atwood, Phil Mattingly and Jennifer Hansler contributed to this report.