Labour’s shadow universities, science and skills minister said his Conservative counterpart, Greg Clark, provided a £550 million figure for the cost of the policy in the House of Commons on 20 November that conflicted with previous figures.

The government has pledged to scrap student number controls from 2015 onwards, which it originally estimated would mean funding an extra 60,000 places a year.

There has also been confusion over how the policy will be funded. George Osborne, the chancellor, originally said in his 2013 autumn statement that the policy would be funded by the sale of pre-2012 student loans.

However, amid speculation about whether such a sale is possible, government officials have indicated the Treasury is committed to funding the policy regardless of any loan sale.

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Mr Byrne asked Mr Clark to “tell the House now how he will pay for lifting the ceiling on student numbers this year”.

Mr Clark answered: “I will give the right honourable gentleman the answer to his question on how the removal of the cap is being paid for. The Treasury has allocated £550 million to pay for it, and it is fully funded.”

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It is unclear what figures Mr Clark was referring to. The Treasury’s autumn statement document gives a cost of £600 million for 2015-16, and a total cost of £5.4 billion across the four years from 2015-16 to 2018-19.

Mr Byrne followed up with a letter to Mr Clark saying that according to the Treasury scorecard in the autumn statement, “the total allocation for this policy is £120m in 2014-15 and £290m in 2015-16, making a total of £410m”.

He adds: “I would be grateful if you could give me an urgent explanation of the £140m difference between the total figure in the Autumn Statement and the figure you gave…as it would be unfortunate if this statement to the House was incorrect.”

But it also unclear which figures Mr Byrne was referring to.

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A statement from Mr Byrne’s office said that figures from the House of Commons Library “suggest a big cut in the funding from £8,000 to £4,000 per place if the government proceeds with plans to uncap student number controls”.

The shadow minister said that whatever the explanation for the different funding figures “it may mask a huge cut in funding for student places revealing the government is indeed short-changing universities for a plan made up on the hoof. It looks like utter chaos.”

john.morgan@tesglobal.com