The petition, called "Save Our Statues, Save our Square" and collected in November and December, accuses the council of paying lip service to a public consultation and compares the statuary to the collection at civic museums.
It also says the category A-listing of each individual statue is an acknowledgement of their international importance.
The plans for the £15 million revamp are currently in disarray, with a decision on the winning design scheduled for next week. However, senior council insiders have cast doubt on the prospect of any of the six being selected.
The council expects to remove all of the statues in due course, at least temporarily.
Campaigner Ann Laird said: "The petition signatories, angry at the staggering lip- service to consultation, are calling for the council to hold a public meeting.
"Scottish planning legislation requires a 12-week pre-application consultation about proposals for a block of 50 flats.
"Where large or significant developments don't happen to qualify for this, developers are advised to hold a voluntary consultation, fully open to the public. Sadly, it seems the city doesn't follow its own advice."
Fellow campaigner Linda Mawson said "We are not dinosaurs and we do want to see George Square improved, but let's do it in a properly considered manner.
"This petition has been handed into the City Chambers simply for the attention of council leader Gordon Matheson."
Green councillor Martha Wardrop: "At a recent committee meeting I raised a series of questions and requested a six-week public consultation before the design brief was finalised.
"This request feel on deaf ears but I hope this can be agreed as a way forward."