Sir Keir Starmer offered to reduce tuition fees for European students coming to Britain as part of an agreement on closer economic ties with the bloc.
The prime minister was engaged in discussions with top EU officials earlier this month as he attended the G7 summit in France.
A deal could see EU students have their fees cut from £38,000 to £9,535, bringing them in line with UK undergraduates, with reports suggesting the reduction was offered as a price to secure a date for a UK-EU summit.
But it is understood the prime minister was clear he would only consider the move on tuition fees if it was part of a broader discussion on the future of Britain’s relationship with the EU.
A UK government spokesperson said: “We do not recognise these claims. We are focused on building a closer relationship with Europe that works for the British people.
“We will not give a running commentary on ongoing talks.”
A date for the second EU-UK summit was set to take place in Brussels on July 22 following the G7 summit, paving the way for the government to secure deals on a youth mobility scheme.
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But the meeting was postponed after Sir Keir announced his resignation just one week later, with Andy Burnham now poised to become the next prime minister.
Before Brexit, EU students were charged the same as British students to attend UK universities, but have since been made to pay international fees of up to £38,000.
University fees are said to be one of the key sticking points in negotiations between the UK and the EU on a youth mobility scheme.
Sir Keir had put a “reset” of relations with the European Union at the heart of his premiership before he announced his resignation.
The outgoing prime minister previously vowed not to return to freedom of movement, a customs union, or the EU single market in what he billed as his ‘red lines’, but it is unclear what Mr Burnham’s approach to the UK’s relationship with the EU will be.
Mr Burnham previously said he would like to see Britain rejoin the EU “in my lifetime”, but later toned down his position he as he fought a by-election in the pro-Leave seat of Makerfield.
Former PM John Major says the UK needs to rejoin the EU’s single market within five years as a matter of priority and cites a loss of £100bn a year in trade and £40bn in tax revenue.
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