Budget cuts for Bristol for the next three years can be reduced from £90m to £83m, the city's mayor has said.
The extra money has come from more tax being collected than expected and government funding changes.
George Ferguson will update the council's cabinet on 16 January as to how he now wants the cuts to be made.
But he said a proposed council tax rise of 1.95% would not be worthwhile if central government decides it has to pay for a referendum on it.
Mr Ferguson said his proposals for council services would take into account the results of a public consultation.
But he said final plans could not be made until after 30 January when central government decides the threshold for a referendum on a council tax rise.
Public toilet provision
"It's very unsatisfactory and very last minute," Mr Ferguson said.
"If they lower the threshold to 1.5% [from 2%] it becomes pretty marginal whether it's worthwhile charging any extra council tax."
About 4,000 people responded to a consultation on the initial budget proposals - 12 times more people than responded to a consultation in 2012.
Of those, 69% were in favour of a council tax rise of 2% or higher.
The main proposals attracting disagreement in the public consultation were cuts affecting older people, including the end of warden and alarm services in independent housing schemes.
Proposals to cut public toilet provision also attracted opposition.
The most support was for reducing the running cost of council buildings and increasing mooring charges at City Docks.
Mr Ferguson said that as an Independent he was "a party of one" and would still need to convince the four other parties of his latest proposals.
'Work to do'
Deputy mayor Geoff Gollop, a Conservative councillor, said extra money in the budget meant "more wriggle room" to avoid making the "most unpleasant savings".
But he said: "A referendum [on council tax] costs money and uncertainty of the outcome means we'll have to consider very seriously whether it's a practical route to go down."
Tim Kent, leader of the council's Liberal Democrat group, said the new figure of £83m was "closer to the £80m we were always saying was the figure we needed to be looking for".
He added: "We still think there's more money that can be found."
Helen Holland, leader of the Labour group, said she did not think the consultation had heard from "the people who will be most impacted".
"There's still a lot of work to do and we've got a couple of weeks to work up our amendment."
The full council will meet to consider the proposals on 18 February.
Full details of the planned budget cuts can be found the Bristol City Council website.