Washington (CNN) Senate Majority Leader Mitch McConnell is leveraging his political power to try to convince Arizona Gov. Doug Ducey to choose Rep. Martha McSally to replace Republican Sen. Jon Kyl, who is considering leaving his seat before his term ends, several Republican sources familiar with the conversations tell CNN.

McSally, a Republican congresswoman who once worked for Kyl, lost her Senate bid two weeks ago to Democratic Rep. Kyrsten Sinema. Kyl, who was picked by the governor in September to fill the seat vacated by the late Sen. John McCain. told CNN last week he has decided whether to leave office before his term ends at the end of next year. He wouldn't reveal his decision, but said he will talk to Ducey about it.

CNN has since learned that McConnell and other national Republican powerhouses are asking Ducey to name McSally as Kyl's replacement, according to several Republican sources familiar with the conversations. If chosen, McSally would fill the position through the end of 2020, at which point the state would hold a special election to fill the remaining two years of McCain's term. The seat would then be on the ballot in 2022 to be filled for the following six years.

While speaking about McSally last week, Kyl told CNN that "Martha McSally would be a very good member of the United States Senate, however she got there," adding, "I regret that she didn't make it in her election."

"I can't think of anybody more qualified than Martha McSally," he said.

Read More