St Andrews University, in Fife, called for a much wider societal effort to address the issue of participation, rather than focusing on the efforts of universities.
The university – where Prince William studied – issued the warning after publishing its outcome agreement with the Scottish Funding Council, which details its future priorities.
Announcing the new target of 20 students, St Andrews said only a small proportion of pupils from poor backgrounds get the grades required for higher education.
And the university said attracting more students from poorer backgrounds would risk lowering its academic standards significantly – stating that students with any less than 3 As would struggle to pass exams and achieve a degree.
Stephen Magee, vice-principal with responsibility for admissions at St Andrews, said: "We have a choice – we can continue to beat up our leading universities for failing to admit more kids from our most deprived areas, or we can start, without shame or blame, to ask if perhaps there is something going wrong throughout the whole equation."
Mr Magee said the university ran a range of initiatives to widen access such as contextualised admissions and bursary and scholarship schemes.
He added: "We know that we could play the political game and change these figures overnight by lowering our entry grades, but experience tells us we would simply be admitting these kids to fail, and that would be utterly dishonest.
"If, as a nation, we continue to lay responsibility for widening access solely at the door of universities, the challenge will never be properly met. We believe it requires a concerted effort on health, employment, better housing and a stronger culture of attainment at all levels of education to equip young people with the grades they need to gain entry and to succeed at university."
St Andrews has been under fire since the summer when The Herald revealed it accepted just 14 students from the most deprived backgrounds last year.
Of the 8872 fifth year pupils from the country's most deprived areas in 2011, only 220 achieved three A Higher passes or better.
Of these, 55 included St Andrews as a choice on their UCAS application. The university made offers to 34 and 14 accepted.
NUS Scotland, which represents students, said the aspirations of the university to widen access were "miniscule".
Robin Parker, the organisation's president, said: "Schools and wider society have a huge role to play, but that's no excuse for universities to abdicate their own responsibilities.
"There is too often an implication that to widen access, you must lower standards, but this is a false choice. The evidence entirely disputes this, with students from poorer backgrounds, even with lower grades, matching and often outperforming students from richer backgrounds at university."
A spokesman for UCU Scotland, which represents lecturers, said: "The university should take account of more than just grades when admitting students by assessing the potential of pupils to succeed based on their background."
Neil Findlay, learning and skills spokesman for the Scottish Labour Party, described the new target as a "shocking figure".
He added: "St Andrew's needs to take real practical steps to improve access for all."
However, Liz Smith, education spokeswoman for the Scottish Conservatives, said. "The real focus of the policy should be on raising standards and aspiration in every school."
A Scottish Government spokesman said: "We have already taken action to ensure access to university is based on ability to learn not the ability to pay by abolishing tuition fees, which is delivering record numbers of students in higher education."