In a telephone call confirmed by two people familiar with the conversation, McConnell (R-Ky.) told Ducey: If there is an opening, consider appointing Martha ­McSally, the Republican congresswoman who came up short in her bid for Arizona’s other Senate seat this year.

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But some Republicans in Arizona, including those in Ducey’s inner circle, have reservations about appointing McSally. They have questioned her campaign’s strategic decisions and wondered why she was not able to win in a state that President Trump carried in 2016 and where Ducey coasted to reelection this year.

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Dan Eberhart, an Arizona-based GOP donor who raised money for McSally, said Ducey should take a “hard look” at other possibilities after her defeat. “McSally’s recent but narrow loss is not a strong reason to choose her,” Eberhart said.

The dueling pressures have injected uncertainty into one of the most pivotal decisions Ducey could face as governor. Arizona is an emerging battleground in the presidential and Senate elections in 2020, loading his potential choice with ramifications throughout the ballot.

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On the call with Ducey, McConnell said McSally would make a great senator and noted there was a lot of support for her in the party. Ducey listened but made no commitments, according to the people familiar with the conversation, who spoke on the condition of anonymity to describe a discussion that was not publicized.

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Facing a more challenging Senate map in 2020 than they did this year, Republican leaders are eager to lock down a strong Arizona contender who can represent the state and effectively use the power of incumbency to run a winning campaign in 2020. Many say ­McSally is the best choice.

“I hate to lose Senator Kyl, but if we have to, I think that’d be a pretty good idea,” said Sen. John Cornyn (R-Tex.), McConnell’s top deputy. “But there’s one guy who’s going to make that decision, and his name is Governor Ducey. And I haven’t heard anything about that.”

Ducey has kept his thoughts private and limited his discussions to a small clutch of advisers. In public, he and his team are not even entertaining the idea of an appointment.

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“The governor is hopeful that Senator Kyl will continue to serve in the Senate seat through 2020,” said his spokesman Daniel Ruiz. “We will not speculate on a vacancy that does not exist.”

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Nevertheless, the possible vacancy has captured the interest of Republicans in Arizona and Washington, who have traded flurries of text messages and other communications about it in recent weeks.

One name that has surfaced as a potential appointee in private conversations is Kirk Adams, a former state House speaker who announced Monday that he was stepping down as Ducey’s chief of staff. Adams said he was looking forward to returning to the private sector. Some Republicans said they still believed he could be appointed to the Senate.

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Anyone appointed to the seat this year would face a grueling next few years. Under state law, the seat will go before voters again in 2022.

Kyl said this week that he has thought about his future in the Senate but has not decided what he will do. “I’m not going to discuss it with anybody except the governor,” Kyl said Monday. “After that, I’ll talk about it publicly.” He called McSally “an obvious potential choice” to succeed him but declined to elaborate.

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McSally lost to Democratic Rep. Kyrsten Sinema by fewer than 60,000 votes out of more than 2.3 million cast, or about two percentage points. Ducey defeated his Democratic challenger by about 14 percentage points.

A former Trump critic, McSally aligned herself more closely with the president during a long primary battle in which she fended off challengers running to her right. Sinema, a former Green Party activist, ran as a centrist Democrat.

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Since the contest ended, Republicans in Washington and Arizona have expressed considerable disappointment in the outcome. They have voiced a range of complaints, including that McSally’s team did not more aggressively seize on opposition research about Sinema’s past to brand her an extremist and that she distanced herself too much from ­McCain in a state where many revere him.

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In a post-election memo McSally’s campaign strategists provided to The Washington Post on Tuesday, they wrote that McSally “was continuously running at a disadvantage going into the general election” and cited factors such as robust Democratic spending and a bruising GOP primary.

“While true that Ducey outperformed McSally, Governor Ducey was an incumbent with two successful statewide races under his belt. Ducey ‘s opponent ran as a progressive and the opponent was essentially abandoned by national Democrats,” the memo says.

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Sinema’s win will make her Arizona’s first Democratic senator in three decades. Her victory has generated excitement in the party. Already, there is talk about who might run for the Senate in 2020.

Democrats pointed to several possibilities, including Mark Kelly, a former astronaut and husband of former congresswoman Gabrielle Giffords; Rep. Ruben Gallego; and Grant Woods, a former chief of staff to McCain who served as state attorney general.

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Woods, a Trump critic, said he recently became a Democrat and has moved closer to a run. “We’ve got an exploratory plan in place, and we’re executing it,” he said.

Woods has also started thinking about which Republican he might be trying to unseat and previewed a possible line of attack against McSally.

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“The governor’s unusual in that he used to run an ice cream company and yet he always chooses vanilla,” said Woods, referring to Ducey’s time as the head of Cold Stone Creamery. “This would be your typical Washington, D.C.-influenced cynical pick, in my view. I think it would be a very poor pick, and I hope he doesn’t do it. And the reason is, the voters have spoken. She had her chance, and she lost.”

Some Republicans have noted that Kyl’s appointment was somewhat surprising and that another unexpected twist might be coming up — including the possibility of Kyl staying on longer than many are anticipating.