A referendum could be held next year to decide whether Bristol should continue to have an elected mayor.
The Bristol Liberal Democrats put forward the motion which is being discussed at a full council meeting today (7 December).
If passed, the referendum will be held at the earliest opportunity - in May 2022 - where the people of Bristol would be asked if they want to continue with the current mayoral system or switch to a committee system.
The current system was decided by a referendum back in 2012.
The position is currently held by Labour's Marvin Rees, after he was re-elected as Mayor in May this year.
Councillor Jos Clark, leader of the Liberal Democrat group - who put the motion forward - described the current system as a "mayoral mess".
Cllr Clark - who serves Brislington West - says she wants to see the committee system as it "allows equal responsibility" for decisions being made in the city, rather than power being held centrally.
She said: "This is not about a party or individual. This is about going to the city and saying 'we've had a go at 10 years of the mayoral system, would you like to change it?'"
Councillor Heather Mack, leader of the Green Councillor group also wants to see the committee system put in place. She said: "This city needs more democracy. It needs the voices of all 70 councillors heard in our decisions.
"At the moment we have one person who makes a lot of decisions and that's not properly accountable to all the people of Bristol.
"There are 10 public meetings a year where the Mayor gets asked questions from the members of public - whereas us councillors are in our communities hearing the issues and concerns and responding to people's worries day in, day out."
George Ferguson, who was elected as Bristol's first Mayor in 2012 - told ITV News West Country he felt the current system is a "confused" one, and he wants to see a council leader structure with a cross-party cabinet.
He said: "We now have a Metro Mayor and we are left with competing Mayors that are undermining each other. I'd prefer that they went to a leader and cabinet system but a cabinet that is representative of the voting across the city.
"We've got a party system that means that the parties dominate and I don't think that's healthy."
In 2016, the West of England Metro Mayor was introduced to the areas of Bristol, South Gloucestershire, Bath and North East Somerset.
Labour's Dan Norris holds the position and says he wants to see the Bristol Mayoral role stay.
He said while it "isn't perfect", it is the "best system" the city has had to date.
He said: "Decisions are hard things to make. That's why we have politicians to do that.
"We should be really carefully giving up something that is better than what we've had previously."
He added: "I think there's room for two mayors but I think there's room for debate about what the names of both mayors should be because I do think it does cause confusion.
"We must be careful when we make this decision that we make a good choice so we can get on with making decisions that help the community because that's what democracy is about."