Pompeo, who served four terms in the House and was Trump's CIA director before moving to the State Department, avoided the subject altogether in a tweet he sent following Trump's comments.
The decision by longtime Republican Sen. Pat Roberts to retire has prompted an effort by other GOP senators to recruit Pompeo for 2020. He's said that push has included a call from Senate Majority Leader Mitch McConnell of Kentucky.
Trump tells CBS' "Face the Nation" that McConnell may have spoken to Pompeo but "I asked him the question the other day. He says he's absolutely not leaving. I don't think he'd do that. And he doesn't want to be lame duck."
Pompeo tweeted that it's true he loves serving as "America's most senior diplomat. I love it because I get to lead the great men and women of (the State Department) who bust their tails every day to proudly deliver American values and American diplomacy to every corner of the world. #Swagger."