A council is planning the largest single displacement of poor people from London in the wake of the coalition government's controversial welfare reforms, singling out more than 700 families to be moved up to 200 miles away.

Camden council said that it would shortly be contacting 761 households, comprising 2,816 adults and children, because the coalition's benefit cap – which limits total welfare payments to £500 a week for families – will mean that they will be unable to afford their current accommodation or any other home in the south-east.

The Labour-controlled council warns that the majority of these families have three children and, once the cap is imposed this summer, will need to find on average an additional £90 a week for rent to remain in their homes – which means "sadly the only long-term solution for some households will be to move".

The local authority says it has been forced to look as far afield as Bradford, Birmingham and Leicester and warns that 900 schoolchildren – more than one child for each class as an average across the borough's schools – face having their education disrupted by the move.

Camden's admission that it is prepared to think the unthinkable and shift large numbers of people out of the capital will be a reality check for ministers who have in the past claimed no one will have to move as a result of coalition cuts.

Other London councils have warned about the numbers of families that will be affected by the introduction of the household benefit cap.

Last month Westminster, a Conservative-run borough, estimated 2,327 households would be affected. In Haringey, one of the four councils chosen to test the changes in April, "temporary accommodation" teams are beginning to collect information about the "income, employment status, personal circumstances and household composition" of 1,000 families who may, according to papers seen by the Guardian, have to move to "lower-cost areas outside of London".

Some authorities have also looked at buying properties outside the south-east. In nearby Brent, where 1,100 households will lose £100 a week after the household cap is introduced, the council has "assessed the costs of procurement in different areas of the country such as the Midlands — including Coventry and Birmingham. We have procured properties so far in Luton, Slough, High Wycombe and Hertfordshire."

Camden says it has been forced to act because the government's policy does not recognise the capital's local circumstances. The borough has the fourth-highest rents in the country. Councillors argue work is no route out of poverty because London has the second-highest childcare costs in the world and house prices are pushed up because Camden's average wage is £37,000, 42% higher than the national average.

The result is that rents of three-bedroom properties in Camden are at least double the government's maximum welfare payment of £340 a week for such properties in north London. Yet three-children families in Camden, said the council, will have a £175 a week limit for housing benefit due to the cap. The local authority says it has no more council housing available – it has a waiting list – so has no alternative but to "explore out-of-borough housing options. The local housing allowance [government welfare subsidy] in Birmingham and Leicester for a three-bed is £127 a week."

One single mother in Camden with four children, all under the age of 10, told the Guardian: "I want to stay where I am for my children's education. What it seems like is the government just want London for the rich. They want to move people on benefits to poor areas." Her rent is £340 for a two-bedroom flat in Camden. When the cap comes into effect, the government will reduce her housing subsidy to £204. This leaves a shortfall of £136. The council has offered to rehouse her in Liverpool.

She said: "Not being given that option to choose where you want to live and where your children go to school isn't fair. The government is taking away people's homes and the places where they've made friends. To think that someone has the power to do that over you … Obviously the government made a lot of mistakes and now everyone is taking the brunt of their mistakes. My children are my priority. If I have to move I will but obviously I'm trying to resolve this."

The leader of Camden council, Sarah Hayward, said: "We are deeply concerned with the continued cuts to welfare benefits and how this will impact on Camden. The very high housing costs in Camden and across London mean that low-income households will find it increasingly hard to find affordable accommodation if they are not in social housing.

"I can guarantee that no vulnerable people will be moved from Camden and we will step up our efforts to engage with those most at risk of losing their homes due to these changes. We will need to set aside additional funding to deal with the fallout of this policy as more people present themselves as homeless but sadly the only long-term solution for some households will be to move."

"Camden is a much better place because of the diverse population that make up its social mix and sadly these changes mean that some low income households will be moved further away from their communities, their jobs and their support networks such as friends and family."

The benefitcap has been dogged by fears of the chaos that might ensue from such a momentous change to welfare policy. Last December ministers conceded that there would be no national roll-out and that changes would only apply to four councils — Haringey, Croydon, Enfield and Bromley – from 1 April and not to everywhere at the same time as widely expected. However last night sources said that because of the fear of a legal challenge, the coalition's welfare minister Lord Freud was planning to bring forward the date at which the rest of country implement the policy.

Stephen Timms, Labour's employment spokesman, said: "This government's incompetence is in danger of creating a cap on benefits that actually ends up costing more than it saves. It is becoming increasingly evident that hapless ministers haven't got the first idea what is going to happen when their changes come in or how much hardship will be caused."

A Department of Work and Pensions spokesperson said: "It's not right that benefit claimants can receive higher incomes than families who are in work. That's why we're introducing a cap on benefits – to restore fairness back into our welfare system while ensuring that support goes to those who need it. Local authorities must consider the individual circumstances of the household and they must absolutely not apply a blanket policy of moving homeless families to different districts."