Some in Mr. Ducey’s orbit would prefer him to appoint Kirk Adams, a former Republican state legislator who until this week was the governor’s chief of staff. Mr. Adams was one of the prospects the governor considered after Mr. McCain’s death, and his departure from Mr. Ducey’s office was widely seen as the precursor to a potential appointment in case Mr. Kyl resigned at the end of this session of Congress.

According to multiple Republicans engaged in the discussions over whom to appoint, Mr. Ducey will make his decision by the end of next week. And he will most likely choose Mr. Adams or Ms. McSally. The only other names mentioned by Arizona Republicans were former Representative John Shadegg and Eileen Klein, the state treasurer.

Mr. Ducey met in person with Ms. McSally in early December and the two have been in touch over the phone since the meeting, according to Republicans briefed on their discussions.

The governor is open to appointing the Tucson lawmaker, but his advisers were angry about a postelection memo Ms. McSally’s campaign drafted that sought to shift blame for her loss. They have made clear they want some assurances from her that she would run a different race and rely on a wider array of advisers in 2020.

Ms. McSally aligned herself closely with Mr. Trump in her campaign against Ms. Sinema, a strategy that could prove even more problematic in a year when he is on top of the ticket.

But Ms. McSally is showing signs that she is willing to address some of the concerns about her: At the urging of Mr. Ducey, she met Friday with Cindy McCain, Mr. McCain’s widow, at Mrs. McCain’s Phoenix-area home. Ms. McSally used the conversation to make amends with Mrs. McCain for distancing herself from Mr. McCain during this year’s primary, when the late senator was in his final months battling brain cancer, according to a person close to the family briefed on the meeting.

The McCain family was deeply irritated at Ms. McSally, who was afraid she would anger hard-line primary voters by aligning herself with Mr. McCain. But Mrs. McCain appreciated the gesture Friday and, while she still believes Ms. McSally is an imperfect option, is not opposed to her appointment, according to the person briefed on her thinking.