A meeting of the full council is due to discuss the petition, organised by city centre businesswoman Debbie Barwick.
She started it in response to issues such as the failed Westgate Towers traffic trial and the executive system of local government, which she argues undermines democracy.
Debbie Barwick with the petition to get rid of leader of the council John Gilbey
It calls on Tory council leader John Gilbey and the other members of the ruling executive to quit.
She said: "This action restricting debate is typical of the way Mr Gilbey operates and is indicative of the general lack of democracy. Ironically this is what the petition is about."
Cllr David Hirst was thrown out of the Conservative group last year after criticising the traffic trial, which saw vehicles banned from passing through the towers' archway.
"This action restricting debate is typical of the way Mr Gilbey operates and is indicative of the general lack of democracy. Ironically this is what the petition is about..." - petition organiser Debbie Barwick
Now sitting as a UKIP councillor, he writes an online blog about Canterbury called The Hirst Comment.
The latest entry reads: "The petition reflects a lack of confidence in the Executive following the ill conceived traffic trial.
"The Hirst Comment believes this action to limit the debate is an insult to the public and certainly to those who signed the petition.
"The council executive would be well advised to allow a proper debate and then consider carefully the way forward as public trust in them appears to have all but evaporated."
Cllr Neil Baker, a former member of the executive, will support the petition if more than half of his constituents urge him to do so.
He said: "I was intrigued to find out if there was any underlying feeling towards this issue because there are a number of vocal people on social media making comments against the executive."
Canterbury City Council leader John Gilbey
The agenda states Lord Mayor Cllr Heather Taylor will ask the council whether it wants to increase the length of time devoted the petition debate.
However, this seems unlikely to be approved with the Conservative group holding 36 of the authority's 50 seats.
Council spokesman Rob Davies said: "Extending the debate beyond 15 minutes will be up to members to decide."
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