Ructions in the government over plans to raise university fees will be forced into the open today when Labour triggers a vote in the House of Commons that could bring about the first rebellion of the coalition.

MPs will debate the plan to raise tuition fees to £9,000 a year as students stage their third and largest national demonstration against the plans. Last night the deputy prime minister, Nick Clegg, wrote to the head of the National Union of Students appealing to students not to distort the debate over fees, saying that many believe wrongly that they will have to pay fees immediately instead of when they graduate. Clegg warned of potentially "tragic" consequences whereby the poorest would be put off applying at all.

Coalition MPs are under a three-line whip to attend the opposition day debate on a Labour-authored motion that falls short of opposing higher fees, but calls for the white paper on the future of universities to properly explain the plans before the Commons votes on the fee cap.

It is carefully worded so that Liberal Democrats in the coalition could vote against the rest of the government without ultimately killing off the planned increase in fees. If they do rebel, it could aid behind-the-scenes efforts to win concessions before the plans are voted on in parliament within the next two weeks.

John Denham, the shadow business secretary, said: "What we're trying to achieve is to show that parliament is being railroaded into a decision just on the fee cap when there are crucial questions on how the policy would work that haven't been answered with a proper white paper."

The Labour move comes amid mounting pressure on all 57 Lib Dems in the coalition. Yesterday a petition signed by 104 former parliamentary candidates for the Lib Dems, essentially representing the party's grassroots, called on Clegg to abide by the pre-election pledge to vote against higher fees. Research published today suggests that the higher fees will profoundly damage social mobility.

One option the leadership is considering is organising a mass abstention to avoid any damaging splits. A party source said that Clegg was still in "listening mode" while also pressing for more concessions and trying to avoid a full-scale rebellion. Clegg's letter to Aaron Porter, president of the NUS, argued that the proposed system was fair because it ends upfront fees for part-time students and raises the repayment threshold to when a graduate is earning £21,000. He warned it would be a "tragedy if we inadvertently allowed our debate about the methods to damage our shared goal" to encourage more students from poorer homes to go to university.

Analysis by the lobbying group representing new universities, Million +, suggests the reforms will limit social mobility and see almost two-thirds of graduates paying much more for a degree than they do now. The "triple whammy" of higher fees, real interest rates for loans and a longer debt write-off period is likely to represent a bad deal for taxpayers and will leave between 60% and 65% of graduates worse off, with middle-income earners hit hardest, according to the study.

Allowing universities to charge £9,000 a year will result in many women spending most of their working lives in debt, while pupils from poor families and mature students will be put off applying, it said.

Million+, which represents 28 former polytechnics, said the findings discredited the coalition's insistence that the moves were progressive. It claims the changes will leave taxpayers worse off because the government will have to borrow more to fund higher loans and pick up a bigger bill for debts written off after 30 years. Write-off costs are likely to rise from 27.5p in the pound to at least 36p, it estimates.