Bristol Mayor George Ferguson will today be urged to continue with a "no evictions" policy over council tenants who fall into arrears because of the so-called "bedroom tax".
Tenants lose £12.70 a week in housing benefit if they are judged to have one bedroom more than they need or £23.90 a week if they have two bedrooms too many.
Since the controversial tax was introduced in April, it is believed that more than 1,000 tenants have fallen into arrears.
The latest council figures suggest that the amount of arrears is now running at more than £750,000.
The issue was postponed from the last council meeting in September because a review was under way and budget proposals were soon to be announced.
Since then, Mr Ferguson has revealed that £90million cuts must be made during the next three years, which will see hundreds of council staff lose their jobs.
Labour Councillor Ron Stone said: "Since the bedroom tax has been in force it has not been clear if it will actually save much money.
"Indeed if we have to evict anyone and make the home fit for a new tenant that will cost around £10,000 each time – which certainly does not make any financial sense.
"About 4,000 tenants are affected and some can pay but the mayor cannot make more than a thousand people homeless – especially as the council would still have a legal duty to rehouse those tenants – so we need a more practical solution."
His motion urges that Mr Ferguson continues with a no-eviction policy for those tenants affected by the Under Occupancy Charge.
The motion will be supported by the Liberal Democrats but they are also seeking help for tenants who make legal challenges against the tax.
Lib Dem leader Tim Kent said: "Tenants shouldn't be left to fight this on their own."