A Los Angeles Times story on Tuesday raised the possibility, saying Brown wouldn't rule it out.
The mere possibility is enough to set political tongues wagging. A presidential campaign in 2016 would be the fourth for Brown, who ran most recently in 1992, when he famously came in second to Bill Clinton. It could also set up a dramatic rematch between Brown and the Clintons, should Hillary Clinton decide to run for president.
Their relationship has been marked by some animosity since the 1992 presidential campaign, which turned ugly between the two men. Brown accused Bill Clinton during a 1992 debate of “funneling money to his wife's law firm for state business,” calling it “corruption.”
In 2009, Bill Clinton retaliated by endorsing Gavin Newsom, then the mayor of San Francisco, over Brown for governor.
But if Brown wants to take on the Clinton machine again, he isn’t yet acting like it.
Much like New York Gov. Andrew Cuomo, another Democrat whispered about as a potential presidential contender, Brown does not often travel outside of his state and keeps his focus primarily at home.
By extension, Brown has not established the kind of national political infrastructure he would need to launch a campaign for president.
"When he has to fly to Washington, he flies Southwest with no staff,” said one national Democratic fundraiser. “He has no national operation.”
He has also not been seriously active within the Democratic Governors Association or in campaigning for other Democratic governors, the usual signs of presidential ambitions.
Brown's job-approval ratings are relatively high, in the mid-50s, but there's another number about which he may be concerned: his age.
The man once derided as "Governor Moonbeam" and linked to the youth movement, Brown is now 75 and acknowledged he is likely past his prime for seeking higher office. The L.A. Times piece quotes Brown at a May event talking about his past presidential primary runs. "Time is kind of running out on that,” he said.
“I guess I’ll just have to stay and do the work of being the governor,” Brown said, “which I actually enjoy because I have some perspective that I didn’t used to have.”