Ministers say keeping the controls in place would be against the law
Ministers do not know how many Romanians and Bulgarians will come to the UK when restrictions are relaxed, the communities secretary has said.
Eric Pickles said he has no confidence in figures, published on his department's own website, predicting about 13,000 will arrive.
The government could only monitor "pull factors" attracting migrants, he said.
Romanians and Bulgarian have faced restrictions on coming to the UK to work since the countries joined the EU.
Speaking at a Parliamentary Press Gallery lunch in Westminster, Mr Pickles said: "I know it's a bloody mistake to tell the truth but... the truth is nobody really knows.
"There is no secret calculation that we are trying to hide from the world” Eric Pickles Communities secretary
"So all the government can do is to just be careful about the pull factors that might range from the health service, through housing, through benefits within the law to try and ensure there isn't an extra attraction to come here.
"Already there are people working in Lincolnshire and our crops being brought in (by people) that come from Romania and Bulgaria and the truth is very few carrots would be picked without that help that is there."Migration prediction
"So there is no secret formula that is too horrible to bring out, there is no secret calculation that we are trying to hide from the world because, indeed the calculation has been on my website undisturbed for the best part of two years and there are other documents that we will be publishing in due course."
The Home Office has said it has not produced forecasts and Downing Street has insisted it would not publish predictions on the number of people who could move to the UK when rules on the movement of people from the EU member states are relaxed at the end of 2013.
The formula on the website suggests 4,613 Bulgarians and 8,155 Romanians would head to the UK.
The countries joined the European Union in 2007 but under "transitional arrangements" their workers were prevented from travelling to the UK.
Migration Watch UK, which campaigns for tougher controls on immigration, estimates 250,000 Romanians and Bulgarians will move to the UK in the first five years after access restrictions are lifted.