Bristol's mayor has said council tenants will not be evicted following changes to housing benefits until "a sustainable way forward" is found.
Labelled a "bedroom tax" by Labour, families deemed to have a spare room will have their housing benefit claims reduced from April.
An estimated 3,700 of the council's 28,000 households are likely to be affected by the changes.
Ministers say the move is needed to cut the £23bn housing benefit bill.
The government wants people to either move to smaller homes if they have a spare bedroom or their benefits will be reduced.
But mayor George Ferguson said there is a lack of smaller council properties in the city for people to move to.
'Outside their control'
"The council will not evict any tenants for arrears they build up due to a genuine inability to pay this new sum until the cross-party working group has had time to examine the issue and propose a sustainable way forward," he said.
"If tenants wanted to move we have nowhere near the number of smaller properties for them to swap into, meaning that coping with this change is often outside their control."
A Bristol City Council cross-party working group is to be set up to examine the issue - including what would happen to the rent arrears that are built up.
Following the mayor's statement to the council, Liberal Democrat councillor Tim Kent said he has written to the Work and Pensions Secretary Iain Duncan Smith calling for the under-occupancy rule to be scrapped.
Mr Kent said: "We welcome this statement (by the mayor), but we continue to have grave concerns. It is bad law and should be withdrawn."
'Not their fault'
Labour councillor Ron Stone added: "People in my ward and elsewhere across the city have been worried sick about losing benefits, and ultimately their home, because they have a 'spare' bedroom.
"It simply is not their own fault that the city cannot find them smaller accommodation to move to once children leave home, or other circumstances change."
Housing benefit claimants typically receive between £50 and £100 a week.
A spokesman for the Department for Work and Pensions said they would not comment on any one council's decision on how to support their tenants through the benefit changes.
He added: "We are giving councils £150m this year so that they can help their vulnerable tenants. In Bristol, the extra money going to the council is £1.1m."