The Home Secretary, seen as a potential Conservative leadership candidate, is said to want a future Conservative government to “move towards zero net student migration” and is calling for a manifesto commitment on the issue.
Universities would be responsible for making sure their students left the country, according to a newspaper report on the proposals.
The Sunday Times said the plan, described as “damaging” by Universities UK, would be “seen as an attempt to burnish her credentials as a future leader”.
The newspaper’s report was based on a briefing from a “source close” to Ms May.
“Theresa is pressing for the next Conservative manifesto to contain a policy that will make sure that anybody coming here on a student visa will have to leave the country in order to apply for a new visa of any kind,” the source said.
“She wants to make the colleges and universities that sponsor foreign students responsible for ensuring their departure. And she wants to be able to fine colleges and universities with low departure rates and deprive the worst of them of their right to sponsor foreign students.”
Nicola Dandridge, Universities UK chief executive, said: “Clamping down on genuine international students would not only damage our universities, but would also damage our economy.
“If the UK is to remain internationally competitive, it should be looking to broaden, not limit, the opportunities for qualified international graduates to stay in the UK to work for a period and contribute to the economy.”
She added: “According to recent ICM polling, the majority of British people (75 per cent) are in favour of allowing international graduates to stay on and work for a period after they finish their degree. Among Conservative voters, support rises to 81 per cent.”
Ms May’s insistence on continuing to include university students in the net migration targets used by the government has long been criticised by the sector.
In 2012, the then University of East Anglia vice-chancellor, Edward Acton, accused Ms May of acting “like a Dalek” and of “casting a dark cloud over British higher education” in her refusal to change course on student visas.