Roma residents of Pavlovce say they would take the chance to leave
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Kent County Council has criticised the government's handling of immigration controls ahead of EU expansion in May.
Britain is one of only two existing EU countries to allow people from states like Lithuania, the Czech Republic and Poland the right to work.
Many people are expected to arrive in the South East, at Dover and Gatwick.
Leader of Kent County Council, Sir Sandy Bruce-Lockhart, said people unable to earn a living will become the responsibility of the authority.
He said: "Our concern is that if we simply have free migration, which seems to be the idea, and the government's answer is not to pay benefits, then these people will fall destitute.
"They will become the responsibility of the local authorities."
Many of the Roma population in Pavlovce, a village in Slovakia where unemployment benefit has been halved this year, say they are suffering from discrimination, lack of money, food and jobs.
Jori Kotlar, one villager who said he would take the chance to leave, told BBC South East Today: "We want to work.
"My profession is as an electrician, but I don't just want a job in my profession.
"I would take cleaning, gardening - any job."
Issues discussed by the government on Tuesday included a possible work permit scheme and restrictions on benefits.
Full details of a package agreed by Tony Blair and key ministers will be given to MPs on Monday.
The government has indicated it will try to stop people trying to tap into UK benefits systems, but enable people to work in the UK's "buoyant labour market".