“In other words, Arpaio doesn’t stand a chance in hell in today’s Arizona,” said Carlos Vélez-Ibáñez, a scholar of border studies. “The guy also has to deal with the same Republicans who voted him out of office in 2016,” he added, referring to the sheriff’s election won by Paul Penzone, a Democrat.

Venomous insults have been lobbed between the Republican campaigns this primary season. Kelli Ward, a hard-right former state senator known for promoting conspiracy theories, is vying with Mr. Arpaio for the support of Trump loyalists, while Martha McSally, a former Air Force fighter pilot, remains in the lead.

Some conservatives here were already critical of the controversies around Mr. Arpaio, if only because Maricopa County has spent tens of millions of dollars for his legal defense in various lawsuits. Others say his aggressive conservative positioning is out of step at a time when appealing to relatively moderate independents is viewed as important in challenging Democrats, who are growing more forceful in Arizona.

Signaling their own assessment of Mr. Arpaio’s chances, Democratic and Independent campaign committees have spent more than $2 million opposing Ms. McSally, the perceived Republican front-runner in the race, according to The Arizona Republic.

Ms. Sinema, running against the lawyer and consultant Deedra Abboud on the Democratic side, is running well ahead in the polls and already is focusing her attention on the fall campaign.

To appeal to voters in the Republican base who often cast ballots in primaries, all three candidates have sought to promote their support of Mr. Trump. Ms. McSally has shifted to the right, becoming more hard-line on immigration, but not as hard-line as Mr. Arpaio, who has said he wants to deploy the Army to Mexico to combat drug smuggling.

“I think he peaked long ago,” Dan Eberhart, a prominent Republican donor in Arizona, said of Mr. Arpaio. “Ward and Arpaio have been in a wrestling match for the far-right vote,” calling the campaigning “unfortunate.” “But their battle clears a path for McSally. At the end of the day, she’s our best chance for keeping the seat.”