Tenants participating in a trial of the government's controversial benefit cap are being sent eviction letters because the welfare changes mean they "may not be able to afford the rent" and they may have to leave their homes within 14 days, according to documents obtained by the Guardian.

In the first tangible proof that the cap would lead to rising levels of homelessness, one of Britain's biggest social landlords, Genesis, has issued a warning to tenants in Haringey – a London borough chosen by ministers to test plans to limit benefit payments to £26,000 a year – saying that it will now need to start legal proceedings to "terminate our lease".

The letter from Genesis says it has been forced into taking these steps because of the "significant changes being currently introduced to the welfare benefit system". The letter warns that, if the tenants do not offer a defence, a court can force eviction within 14 days.

The overall benefit cap set at £500 per week, or £350 for single people, was introduced in four London local authorities – Enfield, Bromley, Croydon and Haringey – in mid-April, and will be rolled out nationally later in the year. The Department for Work and Pensions has estimated that 56,000 households will be hit, with an average weekly loss of £93. The majority will be families with children.

Haringey council says it is "astonished by the premature threat" of eviction – which raised the possibility that scores of families may end up on the streets. It has 660 households who face an average £50-a-week loss because of the cap – and is spending £1.5m over three months to offer them homes within 1.5 miles of the borough.

Claire Kober, the Labour leader of Haringey council, said she knew of a number of social landlords using the threat of eviction since the benefit cap was introduced. "This behaviour of housing associations is completely unacceptable, especially given their stated social mission," she said. "But this underlines to me that the fears that we expressed to government about the consequences of the benefit cap are coming true. The benefit cap is not addressing the cause of the rising housing benefit bill – just the symptoms."

The government, wary of how councils would cope, deliberately slowed the implementation of the benefit cap, with councils applying the cash limits to just 60 claimants a week.

Duncan Shrubsole, policy director at the charity Crisis, said: "What's happening in Haringey is an example of how brutal it is out there. Enfield council is considering moving 330 people out to places like Birmingham and Bradford. Every council testing this benefit cap has teams of housing officers warning tenants that they face some stark choices."

In February, the Guardian revealed that Camden council in north London said 700 families faced being moved up to 200 miles away because the coalition's benefit cap would mean they would be unable to afford their current accommodation or any other home in south-east England.

A spokesperson for Genesis at first denied that letters had been sent out. When confronted with the text of the letter, the social landlord, which manages about 30,000 homes across London and south-east England, said there had been a "cack-handed attempt" to explain the situation in Haringey to "some clients". "That letter should not have been written that way. We are working with tenants in Haringey to help them out of arrears."

Last month, the government was accused of misrepresenting statistics to claim that the cap on benefits had driven 8,000 people to find work.

The Department for Work and Pensions said: "We have been working hard over the last 12 months to support claimants who might be affected by the benefit cap – with Jobcentre Plus, councils and housing associations providing practical help, such as training to get into work and advice over housing options."

The letter available for download has been altered at Genesis' request to protect the privacy of a junior member of staff