Secretary of State Mike Pompeo has told Senate Majority Leader Mitch McConnell that he won't run for Senate in 2020, a source close to McConnell told Fox News on Monday.

Pompeo has been urged by Republicans to consider a run for the open seat in Kansas to replace outgoing Republican Sen. Pat Roberts. Pompeo represented in Congress from 2011 to 2017 before Trump appointed him to lead the CIA and later the State Department.

In recent days, Pompeo himself has stoked speculation that he might run, including by making numerous visits to Kansas in 2019 and creating new personal Twitter and Instagram accounts. The Wall Street Journal noted that Pompeo posted a photo of his meeting with the Kansas Farm Bureau, which holds significant power in the state.

"Leader McConnell spoke with Secretary Pompeo this afternoon where he indicated he will not be running for Senate," the source told Fox News. "Leader McConnell believes Secretary Pompeo is doing an incredible job as Secretary of State and is exactly where the country needs him to be right now.”

Pompeo, who was born in California and moved to Kansas in the 1990s, has time to change his mind, as the filing deadline for the Kansas primary race isn’t until June. He has more than $1 million in a campaign account from his time in the House that he could use.

The development comes as Pompeo has assumed a visible role in promoting President Trump's Iran strategy, following Trump's decision to order the killing of Iranian Gen. Qassem Soleimani.

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Despite numerous overtures, Pompeo has repeatedly suggested he wants to stay in his current job.

“It’s my intention to stay and continue to serve as President Trump’s secretary of state,” Pompeo told Fox News in December, adding: “I’ve said that consistently. ... I’ve watched my life take turns that one would never have expected. But it’s not something I want to do.”

Also in December, White House National Security Adviser Robert O’Brien told ABC News that Pompeo had also waved him off a possible run.

Secretary of State Mike Pompeo, center, Secretary of Defense Mark Esper, left, and Chairman of the Joint Chiefs of Staff Gen. Mark Milley deliver a statement on Iraq and Syria, at President Donald Trump's Mar-a-Lago property, Sunday, Dec. 29, 2019, in Palm Beach, Fla. (AP Photo/ Evan Vucci)

“I spoke with him about this two nights ago,” O’Brien said. “He said he's not running for Senate. He said he's staying as secretary of state.”

At the moment, Kansas Secretary of State Kris Kobach leads the polls in the state’s GOP primary, but Pompeo would be expected to be the frontrunner should he enter the race to replace retiring GOP Sen. Pat Roberts.

Given Pompeo’s popularity in the Sunflower state – and Kobach’s past history of making controversial statements – Republican lawmakers have been rumored to be pushing the secretary of state to run for the seat. Kobach lost his bid in 2018 for Kansas governor.

“Let me put it this way: it’s certainly his decision,” Sen. Todd Young, R-Ind., the National Republican Senatorial Committee chair, told radio host Hugh Hewitt last year. “I can conceive of no one who I’d rather work with in the United States Senate from the state of Kansas than Mike Pompeo.”

Even Pompeo’s current boss, President Trump, has weighed in on a potential Senate run for the secretary of state, saying that Pompeo would “win in a landslide” if he jumped into the race.

After Trump remarked Sunday that Iranian cultural sites were fair game for the U.S. military, Pompeo said the U.S. military may well strike more Iranian leaders if the Islamic Republic retaliates. But, he tip-toed around questions about Trump's threats concerning the cultural sites, a military action that likely would be illegal under the laws of armed conflict and the U.N. charter.

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Pompeo said only that any U.S. military strikes inside Iran would be legal.

“We'll behave inside the system,” Pompeo said. “We always have and we always will.”

Fox News' Rich Edson and Andrew O'Reilly contributed to this report.