Senate Minority Whip Jon Kyl has repeatedly demurred when pushed to reveal whether he will run for re-election in 2012. The Arizona Republican appears amused by rumors that he is considering retirement and perplexed at being questioned about his political plans so soon after this year’s midterms.

Kyl, during a brief interview before Thanksgiving, joked that any announcement about running for a fourth term would be handled to maximize favorable publicity, but then he expressed exasperation with the media’s focus on an election that is two years away.

“When I make my announcement, I want to get some press on it. If I make it now, then I don’t get any press,” Kyl quipped. But then his remarks became serious.

“One of the lectures I give is, what’s wrong with the political system today,” he said. “It’s that every election starts the day after the last one. That is what’s screwing up our system, because there’s no time out to do legislation in a bipartisan way or without the press of elections. And, so I said, I’m not going to be a part of that game. I’m not going to start that right now. I say the same thing when people talk to me about presidential candidates: ‘I’m not going to go there.’”

Should Kyl run, he would begin the race as a heavy favorite. And should he win, he stands a good chance of becoming Majority Whip in the 113th Congress, given the Republicans’ prospects for regaining control of the Senate in 2012.

Republican operatives who monitor the Senate, including one GOP strategist based in Arizona, are split when it comes to speculation on Kyl’s plans.

Some sounded as surprised as Kyl about the rumors that he would call it quits on the brink of ascending to the majority, with one knowledgeable GOP lobbyist saying flatly that Kyl is definitely running. But others aren’t convinced.

Kyl is described by a second Washington, D.C.-based GOP lobbyist as a “lawyer’s lawyer” who runs his office like a well-oiled law firm. The former House Member is thought to covet a top spot in any Republican administration that might assume command of the White House in 2013, particularly the position of Attorney General, this lobbyist said. Some Republicans also think Kyl has long had his eye on receiving an appointment to the Supreme Court.

Despite the uncertainty about the Senator’s future plans, Kyl’s campaign fundraising team is proceeding as though he will run for re-election, according to sources, and the Arizonan hasn’t shown signs of disengaging from the chamber. He is the GOP’s point man on the new Strategic Arms Reduction Treaty that President Barack Obama signed with Russia earlier this year. He also recently helped negotiate the settlement of a lawsuit brought against the federal government by an Arizona Indian tribe, and he is preparing to help lead a strengthened Minority Conference in the next Congress.

But the rumors persist.

“I think he wants something in a Republican administration,” said one GOP strategist based in Washington. “He is getting tired of the Senate. I doubt he runs, but if does, he won’t serve all six years” of his fourth term.

A Republican strategist in Arizona predicted that Kyl would run, although this individual suggested that the speculation that he might retire was not far-fetched.

“While I had heard he didn’t really want to run again, his fundraising team is moving ahead like he is,” the Arizona GOP strategist said. “I’d be willing to bet he will have at least $5 million in cash on hand by the end of next year. Plus, there will be no credible Democratic challenger.”

Kyl reported just under $620,000 in cash on hand at the end of September.

If Kyl were to retire, Republicans viewed as interested in running for his seat include Rep. Jeff Flake (Ariz.); Rep. John Shadegg (Ariz.), who is retiring in December; Mary Peters, transportation secretary under President George W. Bush; and possibly Tempe Mayor Hugh Hallman. Flake is viewed as a potential frontrunner in any competitive GOP primary.

Rep. Gabrielle Giffords (Ariz.) has long been considered as a potential top-tier Democratic candidate given her electoral success in the Republican-leaning 8th district.