Bristol's elected mayor says the local authority is having to make "wicked decisions" to help balance its books.
Marvin Rees says the city council can no longer afford the council tax reduction scheme, a means-tested benefit to help people on low incomes.
The council is asking for views on plans which could see some 25,000 households paying between £300 and £600 more per year.
But campaigners warn it will put even more "on the bread line".
Bristol is one of the last councils to maintain a high level of support for low-income households.
Single mother, Kerry Bailes, said the benefit "has made a massive difference" and warns it could lead to people having to decide whether to "heat or eat".
Mr Rees said the council was "making the best decisions... while at the same time absolutely prioritising our commitment to ending poverty and taking care of the most vulnerable".
"It's one of the perversities of government policies that no-one's really asked: what [are] the cost of the cuts? Short-term savings have long-term costs that are often greater than the savings."
But Ms Bailes from Hartcliffe, believes the move may lead to an increase in homelessness and poverty.
"If this scheme goes people are going to going to be put further onto the bread line.
"Employers aren't going to give people extra wages. You're not going to get any extra benefits to cover it, so it's asking the poorest people in society to pay a debt we didn't create."
The proposal was also condemned by Steve Woodcock, managing director of the St Pauls Advice Centre, who says the proposed cuts will "dramatically affect people on low incomes".