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Mr Cull denied the rumours yesterday, telling the Otago Daily Times he was still considering his plans and expected to make a decision ''in the next month, I would imagine''.
But several other councillors spoken to yesterday - including potential rivals for the mayoral chains - suggested he might be about to step aside.
They pointed to the recent announcement by incumbent councillor Aaron Hawkins, who last week unveiled plans to stand again for a council seat and the mayoralty.
Cr Hawkins had, earlier this year, told the ODT he had no interest in challenging Mr Cull for ''his job''.
Mr Cull said yesterday he had given Cr Hawkins no indication of his plans, while Cr Hawkins said ''whether or not I'm challenging the mayor is a question only he can answer''.
Nominations for October's local body elections do not open until July, but mayoral candidates have already been emerging in other main centres.
In Dunedin, Cr Hawkins remains the only candidate to formally announce a mayoral bid, although Crs Jim O'Malley, Lee Vandervis, Christine Garey and Conrad Stedman have all previously signalled they were considering tilts.
Scout Barbour-Evans and Carmen Houlahan have also confirmed their plans to run as mayoral candidates.
Cr O'Malley said yesterday he was still considering his plans, and would make an announcement within the next two weeks.
Cr Garey was still not prepared to engage in ''speculation and announcements'', which were ''distracting''.
''My current focus, and I mean this most sincerely, is to do my job,'' Cr Garey said.
''There will be a time to make decisions and indicate intentions at a later stage.''
She said speculation was ''rife'', including about Mr Cull's plans, but added: ''Dave has made it clear - he's still making up his mind.''
Cr Andrew Whiley said he, too, was ''still on the fence'' about another bid for the mayoralty, ''but I'm definitely running for council''.
However, he doubted Mr Cull would seek another term.
''I am expecting Dave not to run now ... with Aaron announcing [his mayoral campaign], I figured that was it.''
Cr Vandervis said he was still yet to decide on another mayoral bid, but would stand for a council seat again.
He doubted Mr Cull would run again ''because he realises his chances of winning are poor'', having spent $50,321 only to ''scrape in'' at the last election.
Mr Cull had secured 17,229 votes, down slightly from 2013 but still more than 5000 clear of Cr Vandervis.
Mr Cull said yesterday he still had ''a number of factors to weigh up ... and I genuinely haven't come down on a final decision''.
Those factors included having held the post for three terms, as well as his age - he will turn 70 next year - and family commitments.
There were also significant financial challenges ahead for Dunedin, juggling pressure on rates with the need to invest for growth and climate change mitigation, as well as exciting new developments, and he continued to enjoy the job.
Another issue was whether to seek another term as Local Government New Zealand president, which required him to hold an elected post - either as a mayor or councillor.
Mr Cull said the LGNZ role was a factor, but he looked forward to clarifying his plans soon, ''especially to me''.