Theresa May’s government has been accused of a cover up after it refused to publish the data used to lock Scottish universities out of a new scheme for talented foreign graduates.

Overseas students at four universities in the south of England will be allowed to stay in the UK to look for work after they finish their course.

The Home Office has said that the universities involved, Oxford, Cambridge, Imperial and the university of Bath, had been chosen for their consistently low level of visa refusals.

But ministers refused to publish the statistics, citing commercial sensitivity.

Former First Minister Lord McConnell of Glenscorrodale, who had asked for the information, accused the government of a “deliberate cover up”.

He said he would be pushing for the full publication of the data or a review of the policy.

In her reply to his request, Home Office minister Baroness Williams of Trafford said that the statistics had been produced for internal use only and not for publication.

She added that the information could potentially affect the reputation of the institutions and their ability to attract international students and could not be provided because of "commercial confidentiality".

Lord McConnell said: "It is truly shocking and deeply suspicious that the UK Government has taken this decision using data that they refuse to make public.

"They must have known about this secrecy when they made the decision so this has to be a deliberate cover up.

"Without the data, there is no way to know if the four universities chosen were chosen on merit or because they have friends in high places.

"This is no way to run a visa system and no way to treat the rest of Britain's universities."

He added that while Scotland's universities were being "discriminated against ... ministers are unwilling to even try to prove their case".

Lord McConnell has previously called for the initiative to have a transparent 'threshold' and admit any university with a good record in visa administration.

The two-year pilot was announced by the Conservative Government over the summer.

Under the scheme students can remain in the country for up to six months after they graduate to look for employment.

Universities will be responsible for eligibility checks.

Critics have argued that errors in the visa system are mainly administrative and caused by students or immigration officers, not universities themselves.

Others have complained that the scheme will give some institutions an edge when attracting overseas students and that the south of England does not struggle to attract talented graduates.

Ministers have said that they will consider expanding the drive, and potentially include Scotland, but only after the pilot ends in 2018.

Last month peers accused the Conservative Government of delivering a "slap in the face" to Scottish universities – and warned that the decision could hasten the break up of the UK.

Figures suggest that Scotland stands to lose out on the talents of tens of thousands of graduates, including 3,584 currently studying at Edinburgh University alone.

As First Minister Lord McConnell set up the Fresh Talent scheme as a way to tackle Scotland's "brain drain".

It allowed overseas graduates from Scottish universities to stay in the country for two years after they graduated while seeking employment.

Fresh Talent was subsumed into the UK immigration system in 2008, but post-study work visas were scrapped altogether in 2012 by the then Home Secretary Theresa May.

Scotland's International Development Minister Dr Alasdair Allan and Universities Scotland are among those who have called for has also called on the Home Office to include Scottish universities in the new pilot.