At Mar-a-Lago in Trump's Palm Beach resort, Trump pitched Scott as the party's leader for the Senate at a private meeting.
Trump said to his fellow Floridians, "You should run as Senate majority leader," according to three sources who were present at the meeting and confirmed it to CBS News.
Politico first reported that Scott declined the offer but said to Trump that he was focusing on winning Senate seats in November.
Scott stated, "My sole focus is on winning."
Scott answered CBS News' Robert Costa when he was asked Saturday at CPAC if he would rule out a challenge to McConnel.
CBS News was told by a source that Trump hates McConnell. The former president has repeatedly asked Scott to run against McConnell, even though he's asked several Republican senators directly.
Scott stated earlier this month to CBS News that McConnell would be his party's leader in Senate after the midterm elections.
He said, "I cannot imagine there will be any leader other than Mitch McConnell."