With Republicans holding only a 50-49 majority in the Senate, and a vote on Mr. Kavanaugh looming, the governor thought it was vital to appoint somebody who could immediately step in and represent the state’s interests, according to Arizona Republican officials familiar with the process.

Mr. Ducey and Mr. Trump did speak about the appointment, but the president was not told that the governor had picked Mr. Kyl until Tuesday, according to the officials. The president did not request any specific appointee, signaling only that he would like somebody he could work with. Senator Mitch McConnell, the majority leader and a friend of Mr. Kyl’s, was told last week that his former colleague would be named.

Earlier this year, Mr. Kyl said Mr. Trump’s style was “boorish” and that he was “his own worst enemy.” Asked about those characterizations at the news conference, Mr. Kyl said he stood by his remarks.

Mr. McCain and his family never offered Mr. Ducey any guidance on their preferences for the appointment. Since leaving the Senate, Mr. Kyl has been a lobbyist at Covington & Burling. He said Tuesday that his lobbying work would not impede his ability to serve, and Mr. Ducey’s aides indicated he would sever ties to his clients.

Mr. Kyl’s appointment would be only his latest trip through Washington’s revolving door. He was a lobbyist for an Arizona-based law firm before he won election to the House in the mid-1980s.

Covington had touted Mr. Kyl’s role as the Senate’s “top Republican on the Subcommittee on Taxation and Internal Revenue Service Oversight,” and taxation has been the issue that he lobbies on most frequently, according to data collected by the Center for Responsive Politics.

His client list reflects his establishment pedigree. They include Freeport-McMoRan, the Arizona-based mining company; JW Aluminum, a manufacturer of aluminum products; as well as Georgetown University, and the American Council of Education, which represents colleges and universities.