In the broadcast, which will be aired next week during the Liberal Democrat conference, Mr Clegg admitted it was "a mistake" for his party to claim that it would vote against any rise in tuition fees.
"It was a pledge made with the best of intentions - but we shouldn’t have made a promise we weren’t absolutely sure we could deliver. I shouldn’t have committed to a policy that was so expensive when there was no money around," he said.
But National Union of Students president Liam Burns said that Mr Clegg should have apologised for breaking his pledge, not for making it in the first place.
"The Liberal Democrats' commitment at the 2010 election was to abolish fees, a policy they've had for a decade. It's what they said on the doorstep and in their leaflets, and every one of their MPs signed a pledge to back it up. It's why many students and young people voted for the party," he said.
"It's up to Nick Clegg now to reach out and apologise to those he betrayed by breaking his fees pledge and to make amends by changing his policies."
Deputy Labour leader Harriet Harman accused the deputy prime minister of "crying crocodile tears". She added: "This was not just the small print of his manifesto, this was Nick Clegg's key election promise when he asked people to vote for his party. It is not good enough for him to just brush that promise aside."
Meanwhile, a parody of the broadcast, which sets the deputy prime minister's apology to music, has become a viral hit on the internet. The spoof, produced by satirical website The Poke, is set to be released as a single on iTunes after Mr Clegg granted permission - on the condition that any profits go to Sheffield Children's Hospital.