The Liberal Democrat leader, Vince Cable, has written to the prime minister calling on her to publish evidence on the impact of EU migration that he claims the Home Office has suppressed.

After the Guardian published leaked proposals for post-Brexit immigration policy, May repeated her insistence that unfettered immigration was harmful to low-paid workers.

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“We continue to believe as a government that it’s important to have net migration at sustainable levels – we believe that to be the tens of thousands – because of the impact particularly it has on people at the lower end of the income scale in depressing their wages,” she said at prime minister’s questions on Wednesday.

But Cable, who served as business secretary in the Conservative-Lib Dem coalition government of 2010-15, claims that when May was home secretary, she prevented the publication of a series of reports on the issue, which should now be released.

In a letter sent to May on Thursday, Cable says: “As you will recall, there were reviews, studies and reports into whether the assumption that immigration suppressed UK wages was supported by the evidence. These were shared with the Home Office. They largely showed that this assumption was misplaced, and that EU migration was beneficial to the UK economy and labour market, but the information was never published.

“Parliament will soon consider a fundamental reshaping of the immigration system. In that light, I hope you will agree that it is in the national interest now to release these reports in full.”

He urged the prime minister to ensure that the independent Migration Advisory Committee (MAC) would be given the freedom to produce its own evidence of the impact of foreign workers on the UK economy. “Doubtless colleagues will want to know that the MAC is able to publish at its own discretion and to its own timetable.”

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The leaked paper has reignited tensions within the Tory party about the economic risks of cutting immigration, which flared up earlier this month after new data revealed May had drastically overestimated the proportion of foreign students who overstayed their visas.

Proposals for an immediate post-Brexit crackdown on EU migration set out in the draft paper also sounded alarm bells, including among some Conservatives, about the likely response of the EU27 countries.

A delegation of Conservative MEPs met the prime minister on Thursday afternoon to warn her that the European parliament would not accept an immigration system like that set out by the leaked Home Office paper.

The MEPs, who are split between supporting and opposing Brexit, went to see the prime minister for the first time since the election and some were expected to express concern about the approach of the government towards the negotiations.

Sajjad Karim, a Conservative MEP opposing hard Brexit, said he would raise the unworkability of the immigration proposals with the prime minister.

He told the Guardian: “The European parliament will never accept these proposals and there can be no progress in negotiations. A European parliament resolution will come forward soon and it will make clear that these proposals aren’t getting off the ground. Of course exit in a no-deal situation means the government can go ahead regardless – and it may be good for short-term political consumption in the UK – but will be disastrous in terms of the retaliatory responses thereafter.”

Charles Tannock, another Tory MEP, recently broke ranks to say he was so disgusted about Brexit that he was “ashamed to be British in many ways”. He said on Thursday that EU citizens “should be welcomed and given the respect they deserve for the huge contribution they make to the UK economy”.