Kyl will serve through the remainder of this year and will be able to vote on Kavanaugh's nomination.
Kyl is headed to Washington, D.C. today.
McCain, who served 35 years in Congress, died August 25 after battling brain cancer.
The seat will be up for election again in 2022, the year McCain's term would have ended.
In a story published Wednesday, Woods told the paper he'd discussed the idea with Senate Minority Leader Chuck Schumer of NY and other Democratic senators. The Arizona GOP is sharply divided between a Trumpist majority, and a more "Flake-y" fringe.
Separately, GOP senators decided to make Sen. Jon Kyl to fill the Senate seat for the late Sen.
Kyl's appointment would give Republicans their full contingent of 51 senators. The governor's spokesperson said last week that Ducey would wait until the services for McCain had concluded to select someone to replace the Arizona senator. "I am deeply grateful to Senator Kyl for agreeing to succeed his friend and college of so many years".
Up for reelection in November, Ducey faced the dilemma ensnaring other Republicans in 2018: avoid angering Trump, who has a viselike grip on the party, while placating the GOP establishment epitomized by McCain, who dominated state politics for more than three decades.
Kyl, 76, served with McCain in the Senate and rose to the position of Republican whip, the second-highest leadership post in his party's Senate leadership before retiring in 2013. John McCain's (R-Ariz.) term, a choice praised by Cindy McCain.
"It's a great tribute to John that he is prepared to go back into public service to help the state of Arizona", she said.
While shepherding Supreme Court picks, Kyl has not indicated he's a Trump loyalist, saying in a February interview that the president is a "phenomenon that has to be dealt with".
'I think McCain would be very happy with the pick. "Honors his legacy while putting some major horsepower for Arizona in the seat, at least for now", he said.
Kyl's new stint in the Senate is not likely to last long.
Ducey says he received a lot of advice about naming McCain's successor, but kept coming back to one name: Jon Kyl.
Former senator Jon Kyl (R-Ariz.) eulogized the late Sen. Still, he'll have the chance to cast key votes, including on spending bills to keep the government running and whether to confirm Brett Kavanaugh to serve on the U.S. Supreme Court.
The unusual timing of Kyl's appointment could work out for another Arizona Republican who wants to become a U.S. senator - Rep. Martha McSally. Although Arizona has been a reliably Republican state, demographic trends have raised Democratic hopes of winning there in November and in future statewide elections.