Inside Mr. McConnell’s orbit, strategists calculate that radical elements inside the party can be weeded out and separated from what they see as a larger group that has unwittingly fallen victim to a misinformation about the 2020 election.

It was against this backdrop that Mr. McConnell this month issued a statement that “loony lies and conspiracy theories are a cancer for the Republican party,” in a takedown of Rep. Marjorie Taylor Greene (R., Ga.), who has since sought to distance herself from conspiracy theories.

Mr. McConnell is still fuming over last year’s elections, not just Mr. Trump’s unsubstantiated assertions of a stolen election but also their effects on two Georgia Senate seats, which Republicans lost, costing Mr. McConnell his role as the Senate’s majority leader.

“Georgia was a fiasco,” Mr. McConnell said. “We all know why that occurred.”