Senate GOP leaders have been so dedicated to recruiting Pompeo that McConnell directly urged him to consider it in a recent telephone call, according to two people familiar with the conversation, which has not been previously reported.
By the time Sen. Pat Roberts, R-Kan., announced that he was retiring earlier this month, GOP leaders were already eying Pompeo. In the hours ahead of Roberts' announcement, their allies were talking Pompeo up.
But Trump does not want to lose Pompeo and sees him as his favorite Cabinet member, according to two people familiar with his thinking. It was unclear whether he was aware of the conversation between McConnell and Pompeo. A White House spokesperson did not immediately respond late Thursday to a request for comment.
Pompeo, who has risen through the ranks in Washington from congressman to CIA director and now the head of the State Department, has not ruled out a run publicly.
"Secretary Pompeo is focused on serving the President and keeping Americans safe as the Secretary of State," Pompeo spokesman Robert Palladino said in a statement Thursday.
Privately, Pompeo has left the door open, according to associates, though his attention is on his current job and is not in a rush to reach a verdict, they said.
The people who described the conversation between McConnell and Pompeo and the president's thinking spoke on the condition of anonymity to discuss private talks.
Jumping from a coveted Cabinet position to vie for a spot as a very junior member of Senate would be an unorthodox move - and one some top Republicans remain skeptical Pompeo will ultimately make.
But given all of the twists and turns during the Trump presidency, some Republicans close to both McConnell and Pompeo are far from ready to close the book on the idea.
One person who has been in contact with Pompeo foresaw a situation where Pompeo might want to "parachute out if things get bad." The person spoke on the condition of anonymity to be candid.
Democrats now control the House, giving them the power to issue subpoenas and hold hearings on the Trump administration. Meanwhile, special counsel Robert Mueller III has not completed his investigation into Russian interference in the 2016 election.
"He hasn't closed the door," said one of the people familiar with the phone call, speaking of Pompeo's posture on a Senate bid.
From the perspective of McConnell and other Senate Republican leaders, the appeal of Pompeo is clear - he gives them a high-profile recruit with the ability to head off an potentially expensive and divisive primary for a must-keep GOP seat in a ruby red state.
Given his stature, Kansas political roots and ties to Trump, Pompeo would be heavily favored to win if he entered the race. He served in the House for six years before Trump tapped him to head the CIA. Not long after Rex Tillerson's departure from the secretary of state job last year, Pompeo won confirmation by the Senate to fill the job.
When it comes to personnel and recruiting, Trump and McConnell have not always seen eye to eye.
For example, during Trump's presidential transition, McConnell wanted to recruit Ryan Zinke, then a Montana congressman, to challenge Democratic Sen. Jon Tester. But Trump tapped him to be his Interior secretary.
Tester won reelection last fall, avoiding the fate of four of his Democratic colleagues who lost in red or purple states.
Democrats have not won a Senate seat in Kansas since 1932. But the state elected a Democratic governor in 2018 and several female Republican state lawmakers have switched parties out of frustration with Trump and the current state of the GOP.
Republicans are defending 22 Senate seats in 2020, compared to just 12 for Democrats. While Kansas is not seen as a ripe pickup opportunity for Democrats, it could prove to be fertile ground for messy intraparty battle that could lead to uncertainty in the general election.
Other names that have surfaced as GOP possibilities include former Kansas Secretary of State Kris Kobach, a Trump ally who was defeated in the race for governor last year; Rep. Roger Marshall; former governor Jeff Colyer, whom Kobach defeated in the gubernatorial primary; and American Conservative Union chairman Matt Schlapp.
McConnell, a savvy political strategist focused on retaining or even expanding the GOP's 53-seat Senate majority, is up for reelection himself in 2020. He is seeking a seventh term in a state Trump won comfortably and where he would be heavily favored again in 2020.
McConnell and Trump have had a roller-coaster relationship during the past two years. It turned bad after the failed Republican attempt to repeal and replace the Affordable Care Act. But it improved after the GOP passed a sweeping tax bill.
Tensions flared between Pompeo and Senate Republicans in 2016 when Pompeo stoked talk of a primary challenge by criticizing Sen. Jerry Moran, R-Kan., but the relationship appears to be more cordial now. McConnell called Pompeo a "brilliant, tough-minded leader" when Trump nominated him. And Mary Elizabeth Taylor, the assistant secretary for legislative affairs for the State Department, is a former McConnell aide.