A Wisconsin media outlet is reporting that former Gov. Tommy Thompson (R) will not challenge Sen. Russ Feingold (D) in November.

The Web site of Milwaukee radio station WTMJ reported Wednesday morning that an unnamed source “who is close with Thompson’s family” has said “he believes Thompson’s decision is not to run.”

Though Thompson is pushing back against the report, telling the Associated Press Wednesday that “the only person who knows what I’m going to do is myself and I don’t know what I’m going to do,” a GOP source told Roll Call that Thompson will not run.

Thompson, who served as governor from 1987 to 2001 and then as President George W. Bush’s first Health and Human Services secretary, had been weighing a campaign against Feingold for months and ran about even with the Senator in early polls.

Thompson ran for president in 2008 and has flirted with running for Senate before. He’s been making a good living in the private sector since leaving public life.

Republicans eyeing the seat include real estate developer Terrence Wall, former brewing company executive Dick Leinenkugel and businessman Dave Westlake.