Goff would not rule out an attempt, but said he wanted to be a minister in the next government and win re-election as a Mt Roskill MP.
"I keep hearing this rumour from all sorts of places," Goff said.
"I think it originates with the National Party and suspect it might be being put around mischievously."
Goff said Brown should complete his term. "He made a stuff-up and he's paid pretty dearly for that."
If Brown quit before 2016, Goff said he would not throw his hat in the ring. "I'm a central government politician and I'm hoping to be Minister of Trade and Minister of Defence in the government in six months' time."
Political scientist Dr Bryce Edwards said the smart money was on Brown pulling out next time.
"It's not often that politicians know when to step down but he's just had such strong communication about that," he said. "I assume he'll see the writing on the wall."
Right-wing Auckland politicians allowed John Palino to run unchallenged, but Edwards said they wouldn't do this again.
"I think it's absolutely certain there'll be a strong right-wing contender."
Edwards said it was plausible Goff and Williamson had decided to run for mayor but knew there was no point saying so with a general election coming up.
The public hated "seat-warmers" or noncommittal politicians who caused bothersome by-elections, he said.
Edwards said media personalities were also possible 2016 contenders. Of the media crowd, he said Paul Henry came to mind as a politically-savvy Aucklander.
Public relations boss Matthew Hooton warned against assuming every broadcaster made a good politician. "Former journalists, bless them, don't have a very good record in politics. Pam Corkery springs to mind." Other media personalities to join politics included Brendan Horan and Maggie Barry.
Hooton said Goff and Williamson were admirable politicians but had been around for decades. He understood Williamson wanted to run against Brown using the "National Party brand" last year, but this request was declined.
Henry and Williamson did not return calls. Brown's spokesman dismissed the speculation: "The Mayor is too focused on the job at hand to be worried about what may or may not happen in 2016."
Political punting website iPredict has Brown at 31 per cent likelihood to resign before the next election.