Education Secretary Gavin Williamson was warned in early July that the algorithm used by Ofqual to determine A-levelresults would only be 75 per cent accurate at best.
Former Department for Education Director General Sir Jon Cole told Mr Williamson that the algorithm risked getting hundreds of thousands of results wrong, The Times reported.
In a letter to the education secretary, Sir Jon also wrote that the system would likely advantage small cohorts, such as private schools.
Ofqual's own tests on its algorithm found that it was 60 per cent accurate.
A government source told The Times that, following his meeting with Sir Jon, the education secretary then raised the issue with Ofqual, who reassured him that there would be no problems on results day.
Mr Williamson has previously claimed the scale of the problem only became clear over last weekend.
His U-turn came too late for thousands of A-level students who have already made choices about universities based on the grades they were initially awarded, rather than the new system of using teachers’ assessments.
Schools minister Nick Gibb confirmed on Thursday that he had been warned about the algorithm before results day.
GCSE Results (2020) - In pictures
He told BBC Radio 4's Today programme: “He [Sir John] spoke to me about it and he was concerned about the model and he was concerned that it would disadvantage particularly children from poorer backgrounds.
“And so I called a meeting therefore with the independent regulator, with Ofqual, to discuss in detail those very concerns.”
Mr Gibb said it “certainly was foreseen” that private school pupils could benefit from the use of the algorithm.
"We knew that small cohorts had to rely more on the teacher-assessed grade than on the standardisation process, but that applied to the state sector as much as to the independent sector,” he said.
Despite the disparity in how the algorithm affected rich and poor pupils, Mr Gibb defended the model as "fair" but said it was implemented incorrectly.
He said: “What was always at the forefront of my mind was that no young person from a disadvantaged background would see their grades standardised to a greater extent than other young people.
“There was about a 2 per cent difference, that’s broadly what we saw in the national results last week, in contrast to what we saw in Scotland, where there was a big gap between disadvantaged pupils.
“And that’s because in this country we had more data about the prior attainment of young people that was built into the model."
Mr Gibb added that the model was popular and "widely consulted upon".
“So the model itself was fair, it was very popular, it was widely consulted upon – the problem arose in the way in which the three phases of the application of that model – the historic data of the school, the prior attainment of the cohort of pupils at the school, and then the national standard correction – it’s that element of the application of the model that I think there is a concern.”
The Department for Education has been contacted for comment.