Mr. McCain’s admirers made clear their wishes that Mr. Ducey tap somebody as independent and willing to criticize the president as the departed senator, whose last high-profile moment in public life was to excoriate Mr. Trump’s July meeting with President Vladimir V. Putin of Russia as “one of the most disgraceful performances by an American president.”

“I like senators whose minds weren’t captured,” said Senator Jeff Flake, the state’s other senator, repurposing Mr. Trump’s caustic attack on Mr. McCain’s Vietnam service and appealing to his party to “get away from this personality cult.”

Some of Arizona’s pro-Trump conservatives say that Mr. Ducey’s top priority should be to appoint somebody who is unwaveringly loyal to the president.

“I think it should be a conservative, someone that supports President Trump — I think that should be the main issue,” said Joe Arpaio, the former county sheriff whom Mr. Trump pardoned last year and who is currently running for the seat now held by Mr. Flake, who is not seeking re-election.