Community group It's Out City! launched the Sheffield People’s Petition in August, with the intention of changing the decision-making process at Sheffield Town Hall.
It has until August 24 to gather 20,092 signatures and submit the petition to the council and the group said it was ‘very confident' it would be able to do that.
James Henderson, director of policy, performance and communications at Sheffield Council said that a standalone referendum would cost the authority around £550,000.
He added that if the vote was combined with next year's council and police and crime commissioner elections, it would leave the council with a bill of around £170,000.
Mr Henderson said: “If a valid petition is submitted then we are required to hold a referendum on changing the council’s governance system. Any such referendum must be held on or before the next scheduled local election on May, 7 2020.
“If the referendum is combined with the two other elections (police and crime commissioner and Sheffield Council elections) that are due to be held on the same day, then the additional cost to the council would be in the region of £170,000. If the referendum were held as a standalone election on a different day before May 7, then the cost to the council would be in the region of £550,000.
“The European Union referendum cost approximately the same as the estimate for a standalone governance referendum, but because it was a national referendum, the council was able to claim back most of the cost of running this poll from the Government.
“In contrast, a governance referendum would be classed as a local poll, so the costs of holding one will be fully borne by the council.”
The group is hoping to take action under the Localism Act 2011 to prompt a change from the council’s current ‘strong leader and Cabinet model’ to one where decisions would be taken more by committees, involving more councillors in key decisions.
The legislation requires a petition signed by five per cent of those on the Sheffield Council electoral roll to trigger a referendum, which dropped from 20,956 to 20,092 in February instilling new hope into the campaign.
Anne Barr, of the group, said: “I can’t stress enough that, without exaggerating, around 80 to 85 per cent of people we speak to have some sort of beef when we explain the situation.
“Most don’t know how the council is run and then are amazed when we explain it to them.
“If it goes to a vote, it’s not going to be divisive at all.”
Fylde Borough Council saw a similar change in governance in 2015 following a referendum 12 months previous and West Dorset District Council also made changes following a vote in 2017.
But Ms Barr said the group would ‘much prefer’ the council to change its decision-making process without holding a referendum.
She said: “We would much prefer the council to do it off its own accord because that’s to cost the city far less money.
“We would love it if it would say we are going to do this but until they do we have got to carry on and get the 21,000 signatures.”
For more information visit www.itsoursheffield.co.uk or to sign the petition visit www.ipetitions.com/petition/sheffield-peoples-petition.