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. “I want somebody in there who will protect the back of our president and somebody that will not pursue impeachment if there is impeachment.”

Image Mr. McCain with his wife, Cindy, at a presidential campaign rally in 2008. She is one of several people thought to be under consideration to fill Mr. McCain vacant Senate seat.

Arizona’s primaries are Tuesday, and the race to replace Mr. Flake there has evolved into a contest over who can claim the mantle of the president — not that of the iconic war hero who got remarried to an Arizonan, settled in the state and frequently defied his own party over 35 years in Congress.

The Republican establishment’s favorite, Representative Martha McSally, has linked herself to the president as she tries to fend off far-right challenges from Mr. Arpaio and Kelli Ward, a onetime state senator who won nearly 40 percent of the vote in a 2016 primary against Mr. McCain. To do that, Ms. McSally has avoided any connection to Mr. McCain, a gambit that has infuriated the late senator’s family. (For her part, Ms. Ward suggested in a Friday Facebook post that the McCain family had timed their announcement about the senator stopping his treatment to undermine her campaign.)