The education secretary, Gavin Williamson, has told vice-chancellors the government expects a “deal” on academic standards and admissions in return for extending visas for international students.

Addressing the Universities UK annual conference in Birmingham, Williamson linked the restoration of the two-year post-study work visas for international students, announced earlier this week, with the government’s key concerns, including grade inflation and undergraduate admissions in England.

“The prime minister and I have given you what you asked for, what you wanted most. So I have to ask you for something in return. I see this as a deal, a deal between us and you,” Williamson told the assembled vice-chancellors.

Williamson said he wanted universities to tackle potential threats to their reputation as world-leading institutions, including the rising proportion of students being awarded first or upper-second class undergraduate degrees.

“Grade inflation has become even more entrenched. When I was at university, you could count the number of students on my course who got firsts on one hand. I am sad to report that I wasn’t one of them,” Williamson said.

According to a transcript of Williamson’s speech published by the Department for Education, the education secretary also said of first class degrees: “Now it seems that they are 10 a penny.” But Williamson did not use the comment to the delegates.

Williamson continued: “In 1997 – which is when I graduated – 50% of students gained a first or a 2.1. Last year 80% of students did so.”

A spokesperson for Universities UK said: “It is important not to forget that students working harder, improvements in teaching, investment in academic support and widening participation initiatives are also all leading to legitimate grade improvement and such improvements are to be celebrated.”

But within the sector there is disquiet among some senior figures that the steep increases are indefensible.

A report by the Office for Students (OfS), the higher education regulator for England, found that 90% of universities had a statistically significant increase in the proportion of first-class degrees awarded between 2010 and 2018 that could not be explained.

Williamson also attacked the use of unconditional offers, where universities in England offer places to applicants regardless of their exam results, a practice that has grown rapidly since the government removed the cap on student numbers in 2015.

With unconditional offers rising from 3,000 in 2013 to nearly 76,000 this year, Williamson said the higher education sector needed “to get its house in order”. He suggested limiting the proportion of students given unconditional offers, or setting minimum predicted grade thresholds, as a way of cutting the numbers.

“We need to work together on some of these issues. If we don’t tackle them, your hard-won reputation for excellence will be undermined,” Williamson said.

“Worse still, there is a risk that employers will begin to lose faith in grades and foreign students will think twice about investing their time and money in studying here.”

Early next week Williamson will write to the OfS setting out his concerns in a wide number of other areas.

Speaking to journalists, Williamson also criticised the use of so-called “conditional unconditional offers”, in which universities make the offer of a guaranteed place if the applicants make that university their firm choice.

“They sometimes limit youngsters from the most disadvantaged backgrounds from actually going to the very best academic institution that they possibly could do, and I don’t want see that at all,” Williamson said.

On tuition fees, Williamson told journalists the government intended to make a formal response to the recent review by Philip Augar “by the end of winter”. He refused to be drawn on whether cuts to tuition fees would be included in a Conservative manifesto.

Prof Julia Buckingham, vice-chancellor of Brunel University and president of UUK, said the organisation would call for the return of maintenance grants in its pre-election lobbying of political parties.

Buckingham said Brexit remained a significant concern for universities. “There is no shying away from the fact that no deal – whether this is in 49 days’ time or further down the line – is not what our sector wants. The risks and challenges will be significant and could impact the sector for decades to come,” she said.