The prime minister¡¯s advice, revealed in a letter between their two offices obtained by The Times, may signal Mr Cameron¡¯s awareness that the government¡¯s immigration policy has had a negative impact on universities.

The letter was sent by Simon Case, the prime minister¡¯s private secretary, to Holly White, the universities and science minister¡¯s senior private secretary.

Mr Cameron¡¯s apparent advice to Mr Willetts in the letter to help open up academies and free schools to fee-paying overseas students was reported today by The Times.

However, the full letter also contains advice on ÖйúAƬ policy not reported by the newspaper.

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¡°We should¡­seek to encourage more foreign universities to set up in the UK ¨C as the prime minister has discussed with your minister on a number of occasions,¡± Mr Case tells Ms White.

Such a policy direction could raise anxiety among UK universities, who are likely to fear that foreign universities on their doorstep will bring increased competition for overseas and home students.

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On international students, the letter says that ¡°we need to pitch our offer to international students more persuasively. Is there a way we can bring together the ¡®UK offer¡¯ into an easily understood package?¡±

The letter urges Mr Willetts¡¯ office to work with the Home Office ¡°to identify targeted ways in which we can use our student visa offer to attract the brightest and the best¡±.

¡°Options could include, for example, giving PhD students in STEM subjects only a three-year wait before they can apply for citizenship rather than the normal five years,¡± the letter says.

john.morgan@tsleducation.com