A faction of conservative activists, including President Trump’s former chief strategist, Stephen K. Bannon, now hopes to replicate that kind of grassroots-fueled victory across the country. But their targets are not where the rest of the Republican Party is aiming — states with Democratic senators that Mr. Trump won where there are large pockets of conservative voters — but in seven states where Republicans already hold a Senate seat.

Many Republicans consider this strategy suicidal.

“This is the reason that Republicans didn’t regain the majority in 2010 and 2012,” said Josh Holmes, a former aide to Mr. McConnell who is president of Cavalry Strategies in Washington. “You had this for-profit, quote, unquote conservative crowd who found it much more entrepreneurial to challenge from within than to fight back against Democrats. And ultimately what happened in those two cycles is that Democrats won. Republicans exhausted their resources.”

Using attacks on Republican leaders like Mr. McConnell and Speaker Paul D. Ryan as a way to motivate voters on the right — and to raise money — is nothing new for groups that have tried for years to push Republicans to toe a harder, less compromising line. But to make such a senior leader of their own party the public face of a national campaign to unseat fellow Republicans represents a new level of hostility.

As Mr. Bannon looks to recruit Republicans to run in those seven states where Republican seats are up — Arizona, Mississippi, Nebraska, Nevada, Tennessee, Utah and Wyoming — he has asked them whether they will commit to a pledge not to vote for Mr. McConnell as leader. Those who refrain from making the pledge cannot win his endorsement, or the funding source that comes with it.

Many conservatives believe that Mr. Moore won in Alabama in large part because of the harsh criticism he leveled at Mr. McConnell and Republican leaders in Washington.