The government announced plans last year to overhaul the current system, which sees universities make offers on predicted grades.
A consultation has been carried out over this change, which the education secretary has said would support pupils from disadvantaged backgrounds.
Speaking to parliament on Monday, Mr Williamson said the government is currently looking at the results from this and wanted to bring forward a post-qualifications admissions (PQA) system “as rapidly as possible”.
“We would like to do that without legislation and in co-operation with the sector,” he said. “But if we aren’t able to have that co-operation, we will drive this forward.”
The government launched a consultation into this potential reform of the university admissions system amid concerns about the accuracy of predicted grades.
One option being considered by ministers would see students apply to university and receive offers from institutions after A-level results day – and the start of university could be pushed back.
The other option would see students apply in the usual way during term-time, but offers would only be made after results day in the summer.
A Ucas director said earlier this year the university admissions body would “cautiously” support the second model, but the first option – which would see students apply to university after A-level results day and start courses in January – was “a step too far”.
In the consultation document, Mr Williamson said the system using predicted grades is “limiting the aspirations of students before they know what they can achieve”.
“We know that this disproportionately affects the brightest children from the most disadvantaged backgrounds,” he said, adding he wanted to “smash through ceilings” stopping students from reaching their potential.
Speaking about the proposed reform on Monday, he said: “All the evidence from the Sutton Trust and so many others is very clear than PQA goes to help children from the most disadvantaged families more than any other.”
“That is why we will make it happen.”
Last year, a survey by the Sutton Trust, a social mobility charity, suggested working-class students were more likely to say they would have applied to a more selective university if they had known their A-level grades first.
Previous research from the charity found high achievers from disadvantaged backgrounds were more likely to end up achieving better grades than predictions compared to those who are better-off.
Back in 2017, research found the university admissions process relied too heavily on predicted grades and personal statements, which could put poorer pupils at a disadvantage.