Alarming predictions of social unrest and the break-up of civil society have been delivered by the leaders of three of England's biggest cities, amid new evidence that government policies are widening – rather than narrowing – the economic divide between north and south.

In a letter to the Observer, the council leaders of Newcastle, Liverpool and Sheffield – where Nick Clegg is an MP – issue a plea to government to halt cuts that they say unfairly penalise the north relative to the south, before crime and community tensions erupt on the streets.

Their joint assault comes as fresh analysis shows that one of the government's flagship policies to promote housebuilding and economic activity across the whole country, the New Homes Bonus, is draining money away from northern councils, while swelling coffers in their southern equivalents.

Reacting to the latest cuts announced to council budgets this month – under which a further 2% of spending reductions were unveiled by the chancellor, George Osborne, in addition to the 28% already in train – the council chiefs say ministers in Whitehall appear to be adopting a "Dickensian" view of the world, which is a betrayal of traditional one nation conservatism.

Calling for a change in the way funding is allocated, to make it fairer to the north, they say that while everyone agrees on the need for savings, the coalition austerity programme has gone too far and will soon backfire with chaos on the streets and further economic stagnation. "Rising crime, increasing community tension and more problems on our streets will contribute to the break-up of civil society if we do not turn back," they write.

"The one nation Tory brand of conservatism recognised the duty of government to help the country's most deprived in the belief that economic and social responsibility benefited us all.

"The unfairness of the government's cuts is in danger of creating a deeply divided nation. We urge them to stop what they are doing now and listen to our warnings before the forces of social unrest start to smoulder."

The council leaders argue that under the coalition, northern cities have been hit far harder than those in the south, partly because of the withdrawal of government support to deprived areas that was received under Labour. The situation is made still worse, because during economic hard times the demands for services in poorer areas are greater.

Having already budgeted for cuts of hundreds of millions of pounds since 2011, the council leaders – all from Labour-run cities – say the latest cuts have blown further holes in their finances, meaning that by 2018 they fear being unable to provide acceptable levels of essential services including social care for the elderly, refuse collection, and others such as libraries and sports facilities. On top of that they say their attempts to generate local economic activity are running into the sand because of lack of funds.

There are also complaints that the New Homes Bonus, under which local authorities all fund a bonus pot that rewards local authorities in whose areas new homes are built – is sucking money from the north as developers build more homes in the south because of the bigger demand and the higher prices that they command.

Analysis done by the public services union Unison shows that all the top 10 gainers from the scheme in cash terms were in the south while all top 10 net losers were in the north. The figures from the last year show Liverpool was a net loser from the bonus scheme by more than £2m while Basingstoke and Deane gained by £1m. Most of the councils gaining are Tory-run and those losing out are controlled by Labour.

Dave Prentis, general secretary of Unison, said: "It is grossly unfair that the government is using its New Homes Bonus scheme to shift scarce resources from the hard hit north to wealthier parts of the south. Over 60% of money is spread between London, the south-east, east of England and south-west. It cannot be a coincidence that the Tory heartland is where most money is being directed."

The north-south economic divide is a headache for David Cameron's Conservatives, who are desperate to make inroads in the north of England where they must win more seats at the next election if they are to secure a majority. However, it is also emerging as an issue for Labour leader Ed Miliband who, while needing to support his heartlands in the north, knows he has to speak for the whole country if he is to return his party to power in 2015.

In his new year message Miliband specifically addressed the need for his party to speak to southern voters. He said that "one nation Labour is about reaching out to every part of Britain, it's about a party that is as much the party of the private sector as the public sector, a party of south as well as north, a party determined to fight for the future of the United Kingdom, and a party rooted in every community of our land."

A spokesman for the Department of Communities rejected the idea that the north was getting a bad deal. "Every bit of the public sector needs to do its bit to help pay off the inherited budget deficit, including local government which accounts for a quarter of all public spending. This is a fair settlement – fair to north and south, fair to rural and urban areas and fair to shires and metropolitan areas. Newcastle has spending power per household of £2,522 which is over £700 more than the £1,814 per household in Wokingham – Liverpool is receiving £2,623 per household and Sheffield £2,221. Changes to the spending power is close or in line with the national average reduction of just 1.7% per cent, less than last year's figure of 3.3%.

"The government is also helping councils grow their local economies through bespoke city deals, New Homes Bonus, 24 enterprise zones and £770m in Growing Places funding. The New Homes Bonus has provided £1.3bn of funding since 2011 and to suggest it favours councils in the south is wrong.

"In addition we are offering a £450m third year's council tax freeze – potentially worth over £200 to band D residents.