They are scrambling for a modest two-bedroom property on Lickless Avenue.

Over the past two decades, the number of council properties in Leeds has fallen from more than 90,000 to 58,000 because of the right-to-buy legislation and demolitions.

Senior councillors have pledged to build new council homes in Leeds for the first time in nearly 30 years. Earlier this year they gave the go-ahead for a total of 77 homes on three sites in west Leeds – and 27 of the new properties will be council houses.

These will be at Silver Royd Hill and Evelyn Place in Wortley and on the former Waterloo Primary School site in Pudsey.

Council shadow cabinet member for housing Richard Lewis (Lab, Pudsey) said thousands of families are being forced to bid against each other every week in the hope of gaining a suitable home in Leeds.

He said: "Week in and week out I hear of similar cases where huge numbers of people are desperately bidding on a home that they have no chance of getting. The sheer number of people applying each week provides a shocking picture of housing need.

"This demand reflects the state of the housing market and the failure of the private sector to fill the gap by offering reasonably priced rented family housing."

But Coun Les Carter, executive member for housing, said Coun Lewis was failing to provide any solutions.

He said: "The number of people on the waiting list in Leeds is 31,000 households. It would cost approximately £24bn to build the houses to deal with this demand, an impossible sum to achieve without significant government assistance, which has not been forthcoming over the past 10 years.

"Since 1997 the Government has overseen a 64 per cent increase in social housing waiting lists in England. The total number of households waiting for a council home in England stands at almost 1.7m, an increase of over 600,000 on 1997."