“Conventions have a flimsy track record of selecting the most electable candidates,” David Kochel, an Iowa-based Republican strategist, said in an interview on Wednesday. “There’s just no good substitute for a full-scale vetting by a large universe of primary voters.”

Nowhere is the debate over how to limit the influence of the party’s most hard-line activists more intense than in Utah, where in 2010 conservatives cheered as the Tea Party toppled one of the veteran Republican centrists in the Senate, Bob Bennett. The state party’s caucus and convention system helped elevate Mike Lee, a little-known lawyer who replaced Mr. Bennett.

Now Mr. Lee is a patriarch of sorts to Tea Party conservatives in Congress. Though Senator Ted Cruz of Texas was the public face of the government shutdown fight, it was Mr. Lee who largely conceived the strategy in an effort to block funding for President Obama’s health care law.

Back home, Mr. Lee is now at odds with Republican stalwarts like former Gov. Mike Leavitt, who are leading a campaign for a ballot initiative that would shift the state to a primary system where party conventions have no role in picking candidates and, instead, all Republicans can participate.

“If you look at the place where the aggregate heart of the Republican Party is in this state, it is not represented by the aggregate of folks who attend conventions — demographically or ideologically,” Mr. Leavitt said in an interview this week.

The Republican Party in Utah picks its candidates for statewide office, like United States Senate, at a convention of about 4,000 delegates who are chosen at a series of caucuses all over the state.

Supporters of the current system argue that representation is ample on the front end, when thousands of people are involved in selecting convention delegates whom they get to know through the intimacy of local caucuses. Candidates for office, in turn, have to campaign directly to those delegates.