Councils have warned that a £2bn funding pot finally made available to them will not be enough to deliver the new generation of council homes promised by Theresa May. Their message comes as the first ever social housing green paper by a Conservative government is expected to be published this week.

Local Government Association chair, Lord Porter, the Conservative leader of South Holland District Council, Lincolnshire, said he would not be able to build enough council homes to meet local demand because the government was only offering 166 councils extra borrowing capacity and additional funding for social housing.

“I’m building petty cash numbers. I need about 200 a year and I’m not even building 20. It means I’ve got a bunch of people sitting on a waiting list,” he said. “You have to be in the crappiest life circumstances in my area to access a council house. That should not be the case. If you don’t earn a lot of money, you shouldn’t have to rent in the private sector, where rents will be double what they are in the social sector.”

The green paper, still expected before parliament breaks for the summer this week, and May’s pledge at last year’s Conservative conference that she wanted to build a new generation of council homes, have raised hopes that government would free councils to build at scale again.

The latest figures show that social house building has hit a new low, with only 5,900 homes completed in 2017 – the lowest proportion of overall housing supply since records began. In 2011 nearly 40,000 socially rented homes were built in England.

Porter said the Treasury was unlikely to allow councils to borrow against existing housing, termed “sweating”. “We can’t borrow against our own stock, which is insane. We are sitting on hundreds of billions of pounds in assets that are un-sweated,” he said.

Councils have lost around 100,000 socially rented homes since 2012 through right-to-buy sales and conversions to much higher affordable rents, according to research by the Chartered Institute of Housing.

Labour-controlled Nottingham, which completed 180 social homes last year and has one of the best building records in the country, said it had also been refused access to the new funding.

“They have removed that cap for some councils, but not for Nottingham. That makes it very problematic for us to continue the level of building we have been doing,” said Jane Urquhart, Nottingham’s cabinet member for housing.

A Ministry of Housing spokesperson said social housing was a priority and that it would set out its plans in the green paper: “We are investing £9bn in affordable homes and … giving authorities the power to borrow £1bn.”