Mr Davies has promised a referendum on the mayoral system in 2011
Nearly two-thirds of people in Doncaster do not want an elected mayor, a BBC survey has shown.
Of 500 people surveyed, 63% (313) said they would prefer the town to be run by a cabinet of councillors. Only 22% (109) wanted a mayor to be in charge.
And 87% (437) could not even name the current mayor, Peter Davies.
The findings are published as Mr Davies lost a vote of no confidence amid criticism of the way he has run the council since being elected last June.
Councillors voted 36 to 11 in favour of a motion claiming Mr Davies had not been a driving force in taking Doncaster forward and that he had not given adequate leadership to improve the town.
Mr Davies, of the English Democrats, hit back at his opponents, describing them as "political dinosaurs" in an interview with the BBC's Politics Show.
He accused them of "orchestrating" Monday's vote of no confidence to coincide with the start of an Audit Commission inspection because of "serious concerns about the council's performance in the last two years".
In response to the survey by BBC Radio Sheffield, Mr Davies promised the people of Doncaster a referendum on the mayoral system in 2011.
"My evidence is that the people of Doncaster like the mayoral system," he said.
"Michael Portillo came along the other week and we wandered round the market and he interviewed about 30 people and found them all in favour of the mayoral system.
"Incidentally, they were all in favour of me too, which was quite entertaining from my point of view.
"They were all saying that the mayoral system was better than the old days which provided the town with Donnygate and other things we would prefer not to remember."
However, John Kingdom, a politics expert from the University of Sheffield, said the system of electing a mayor could put people off local politics.
"It reduces the role of the ordinary councillor," he said.
"It means ordinary people lose the incentive to stand and serve their community.
"The result is fewer people stand and the ones who do stand are probably what you would describe as lower calibre."