Some 70 per cent of around 2,900 adults questioned said there should be a cap on overseas students, while 22 per cent said there should not.
Before answering the question, respondents were told that 250,000 non-EU students arrive in Britain to study each year, with around a fifth staying on legally after completing their course.
Support for such a cap was strong across the political spectrum, according to respondents' intention to vote, with 70 per cent of Conservative, 66 per cent of Labour and 57 per cent of Liberal Democrat voters in favour of a limit.
The strongest support was in Scotland, at 76 per cent, while the lowest was in London at 62 per cent. Women were more likely than men to favour a limit, with 74 and 67 per cent respectively calling for a cap.
The poll, published by MigrationWatch UK - which claims it is a "non-political body" that is "concerned about the present scale of immigration into the UK" - also found strong support for action against bogus students. Seventy per cent of respondents thought that those found to have insufficient English for their courses should be deported, while 84 per cent said that those found to be working rather than studying should be kicked out of the country.
Almost 90 per cent thought that those who had overstayed their visas after their course had finished should be deported.
Meanwhile, 85 per cent wanted firmer measures to prevent people coming to the UK as students if their real intention was to work.
Sir Andrew Green, chairman of MigrationWatch UK, said: "This gives the lie to those who have been claiming that the public are not concerned about student inflows. When the questions are posed in their factual and policy context the public display the firm common sense that one would expect."
However, responding to the poll, Nicola Dandridge, chief executive of Universities UK, said there were "clear signals" that the British public understood "the contribution overseas students make to our academic and cultural life, as well as to our economy".
"Widely-respected research from the Migration Observatory at the University of Oxford shows that legitimate international students are the visa category people are least concerned about," she said.
"Universities UK's own work in this area has also found that the majority of people agree that UK higher education plays a key role in boosting our country's international competitiveness and is important for the health of our economy.
"We should be encouraging genuine international students to study in the UK, rather than risking our global reputation by sending out false signals that they are not welcome here."
The survey comes after London Metropolitan University had its license to teach foreign students revoked by the UK Boarder Agency for "serious systemic failure" to monitor its overseas students.