Mr. Reed said that the former president’s comments about Republicans not voting “should warrant a sanction from the Republican National Committee.” He warned, “It will impact the 2022 midterms severely.”

The tension between Mr. McConnell and Mr. Trump has escalated lately, as the former president has ratcheted up his attacks on the minority leader, accusing Mr. McConnell of “folding” to Democrats in a recent agreement to raise the nation’s debt ceiling. He called for the leader’s ouster in a recent interview with Fox News, saying, “Mitch is not the guy.”

Mr. Trump’s allies did not deny that he was content to see Republicans pay the ultimate political price for what he and a significant portion of his voters saw as disloyalty.

“President Trump is saying: ‘Hey, I’m putting you guys on notice. My people aren’t coming out,’” said Stephen K. Bannon, the former chief White House strategist, who has been using his podcast to further amplify Mr. Trump’s false claims about the 2020 election. “There could not be a bigger shot across the establishment bow.”

But the harder Mr. Trump’s allies push their election fraud claims, the harder it becomes to satisfy their most hard-core followers. Even Ms. Taylor Greene, who is as far-right and pro-Trump as they come in the Republican Party, cannot seem to always please the fringe. Lately, she has been feuding online with L. Lin Wood, an Atlanta attorney who helped Mr. Trump sow doubts over his loss in Georgia, over which of them truly represents the Trump movement. Mr. Wood has accused her of not doing enough to uncover instances of voter fraud. She has said that Mr. Wood is “not one of us.

Such radicalization comes at a cost, said David Jolly, a former Republican congressman from Florida who is critical of the former president. At a time when the larger political trends are pointing toward trouble for the Democratic Party in 2022, he said, Mr. Trump’s actions risk interfering with what should be a good environment for Republicans.

“This should be a 100 percent, straight-up referendum on Biden,” Mr. Jolly said. “Instead, you have Trump the narcissist trying to inject himself into what should be a glide path for Republicans to an incredibly successful election, by making it all about him.”