Concerns are rising in Bristol that potential students will begin to turn their backs on a university education if tuition fees go up.

The debate was reignited on Tuesday after two surveys showed that leaders of many universities wanted to see a sharp rise in fees.

A body called Universities UK, which represents vice-chancellors, said it favoured a doubling of the current level, which is about £3,000.

Research for the BBC showed university heads calling for an increase to somewhere between £4,000 and £20,000 a year.

Bristol University's vice-chancellor Eric Thomas said two years ago he would like to see fees of £5,000 a year.

Bristol is a member of the Russell Group of elite universities, which is emphasising the need for additional funding for universities if they are to maintain current teaching levels.

But the vice-chancellor of the University of the West of England (UWE), Steve West, said it would be wrong to raise fees in the current economic climate.

He said: "While UWE supports in principle more investment in higher education, it is very concerned about the impact of higher fees on student debt. It is particularly concerned that anxiety about debt will put off those from low-income backgrounds entering higher education."

The National Union of Students (NUS) is concerned about the impact of higher fees and was due to lobby Parliament today. It says students could soon be leaving university with debts of £32,000.

Tobin Webb, president of Bristol University's student union, said some of his members, along with others from UWE and from further education colleges in Bristol would be attending the Commons protest. He said the widening participation agenda – encouraging students from poorer backgrounds to go to university – was in danger of being further damaged by higher fees.

He was glad the issue was being brought to the fore in advance of the Government review of fees which will take place this year.

Mr Webb said: "We believe that the underlying principles of the current higher education funding system are inherently unfair. They are based on the premise that the individual should pay up front because they will benefit in future from having a degree. But higher education also has a public good and is a benefit to employers, so the state and employers should contribute to its cost."

The NUS case has already won the backing of former education secretary David Blunkett.

He said: "While it's clear that no government is going to pull the financial plug on the university sector by simply abolishing fees, it would at this time of global financial downturn, be unacceptable to lift the cap and have a free-for-all across universities."

Liberal Democrat higher education spokesman Stephen Williams, MP for Bristol West, said: "It is no surprise that many of the vice-chancellors involved with this research would like to see tuition fees more than doubled. The conclusions would be very different if students' views were considered.

"The Government must look at ways of easing the student debt burden instead of increasing it. It should publish its fee review before the next election, so all political parties can make their views know to the electorate.

"Liberal Democrats believe that all students should be able to go to university without having to worry about getting saddled with massive tuition fee debt. That's why we think university fees should be scrapped, not doubled."

Higher Education Minister David Lammy said there was an "important debate to be had" about fees.