Dundee University said the problem was particularly acute in so-called Stem subjects such as science, technology engineering and mathematics – which are vital to the country's future economic success.
The university also highlighted reduced school subject choice and rising university entry requirements for the lack of progress in widening access.
The institution will now work more closely with secondaries in the surrounding area to ensure bright pupils with the prospect of going to university develop the necessary skills.
Dundee – which has a better record on widening access than many universities – is also allowing its campus to be used by secondary teachers to teach four Advanced Highers, to ensure pupils from different schools have the necessary subject breadth to apply to university. It already runs summer schools for potential students.
The warning on access is revealed in the university's outcome agreement with the Scottish Funding Council, which sets out the objectives it will seek to achieve in return for its public funding.
The document states: "The disadvantaged nature of our local catchment has meant that, historically, such students have gravitated to us naturally.
"However, data suggests that, more recently, factors such as increases in entry grades and reduced school subject choices, have reduced this flow.
"Summer schools cannot replace years of poor development in mathematical, numeracy and science skills, which makes transition to Stem in higher education especially difficult."
Larry Flanagan, general secretary of the Educational Institute of Scotland, welcomed the university's commitment to address the issue. "Schools across the country work hard to support aspiration among students, but they do so in the face of falling budgets, diminishing resources and declining staffing levels," he said.
"There has certainly been a squeeze on the ability of schools to offer a full range of Advanced Higher classes which relates directly to cuts in both funding and staffing."
This summer, The Herald revealed some universities were recruiting tiny numbers of students from the poorest backgrounds, with St Andrews taking only 13 in 2010/11.
Concern over the lack of progress prompted the Scottish Government to suggest binding targets on access for universities – with the threat of financial penalties for those that fail.
The current set of outcome agreements only apply for one year and do not include specific targets as they are seen as a foundation for future agreements.
All universities have plans to widen access, with a range of strategies being adopted including summer schools, recruitment from colleges and so-called contextualised admissions – where a student's background is considered alongside academic attainment.
However, it is unclear whether any universities will adopt the approach taken by Glasgow University which gives students from poorer backgrounds a place with lower grades if they complete a tough entry course.
The warning from Dundee comes months after Professor Martin McCoustra, chairman in chemical physics at Heriot-Watt University, said his institution had to run top-up maths lessons for first-year students.
Meanwhile, Glasgow University has warned applications from EU students, which have risen after fees were introduced in England, could lead to a reduction in the number of Scottish domiciled students at the university.