David Willetts says in a paper for the Social Market Foundation that the number of places needs to grow from 368,000 to 460,000 to meet demand.
Universities have seen a squeeze on places in recent years, with thousands of students failing to get on courses.
The primary school place shortage could go on to hit undergraduates, he says.
The paper from the universities minister is billed as his projection of the future of higher education in the UK.
It seeks to update a report written 50 years ago when just 5% of young people went to university.
The Robbins report called for and led to a bold expansion in university places. It rested on the guiding principle that higher education should be open to all able and qualified enough to go.
Robbins considered what the demand was likely to be over the next 20 years - based on the number of suitably qualified young people, not economic forecasts of future jobs.
The projections in Mr Willetts's pamphlet follow this model and suggests there will be a need for 92,000 more places for England-based students in UK universities in the next 20 years.
He says the current demographic backdrop is very different to the 1960s, when the country was in the grip of a baby boom, because of the fall in the number of babies born in the 1990s.
But he adds: "However, looking ahead to the 2020s, we can see the increase in the number of births since the turn of the century feeding through into more young people. Those pressures have already been felt in our nurseries and primary schools."
He also argues that as educational standards in England improve over the coming years, there will be a greater supply of young people with the potential to go to university, thus increasing demand still further.
And he goes on to highlight differences between the English higher education system and European ones, hinting at a more liberal approach.
He also says: "The English system is quite unusual amongst advanced countries for having such selective systems of entry to university... Other countries allow virtually automatic progression to your local university if you have the requisite qualifications."
Although the Department for Business, Innovation and Skills has introduced more flexibility in the system, with universities being allowed to recruit unlimited numbers of brighter students (those gaining certain high grades), overall the numbers have remained fairly static since 2010.
But it is expected numbers will begin to recover again this coming year when admissions quotas for individual universities are being relaxed.
When the Robbins report was published in 1963 there were nearly 216,000 full-time students in higher education. Robbins projected this number would grow to 558,000 by 1980-81.
Director of the Social Market Foundation Emran Mian said: "David Willetts provides a compelling case for expanding higher education. This is our best bet for equipping the UK economy with the skills it needs and improving social mobility."
The report stops short of saying how such demand will be met.
The Department for Business, Innovation and Skills said the pamphlet was Mr Willetts's personal view but would not comment further.
The number of places at UK universities will have to grow by at least a quarter over the next 20 years, the universities minister has warned.