"The university will not surrender its independence or relinquish its constitutional rights," Harvard President Alan Garber wrote in a message on Monday to the Harvard community.

The new request from Noem said the institution would lose the "privilege of enrolling foreign students" if it did not comply with the demand for records.

Harvard said it was aware of the new request from Noem, which was made in a letter, the Reuters news agency reported.

International students make up more than 27% of Harvard's enrolment this year. Even before Noem's statement, billions of dollars hung in the balance for the university, after the freeze of some $2.2 bn (£1.7bn) in federal funding.

Trump has also threatened to also remove Harvard's valuable tax exemption, the loss of which could cost Harvard millions of dollars each year. US media reports suggest the Inland Revenue Service (IRS) has started drawing up plans to enact this.

"Harvard can no longer be considered even a decent place of learning, and should not be considered on any list of the World's Great Universities or Colleges," Trump wrote on his Truth Social platform on Wednesday.

"Harvard is a JOKE, teaches Hate and Stupidity, and should no longer receive Federal Funds."

The administration's attacks on Harvard are not isolated. The government's antisemitism task force has identified at least 60 universities for review.

During his presidential campaign, Trump pitched a funding crackdown on universities, painting them as hostile to conservatives. He and Vice-President JD Vance have long railed against higher education institutions.

Polling by Gallup last year suggested that confidence in higher education had been falling over time among Americans of all political backgrounds, particularly Republicans - in part due to a belief that universities push a political agenda.

Since taking office, Trump has focused particularly on universities where pro-Palestinian protests have taken place. Some Jewish students have said they felt unsafe and faced harassment on campus.

In March, Columbia University agreed to several of the administration's demands, after $400m in federal funding was pulled over accusations the university failed to fight antisemitism.

These included replacing the official leading its Middle Eastern, South Asian and African Studies department and pledging to take on a review to "ensure unbiased admission processes".

Harvard too has made concessions - including by dismissing the leaders of its Center for Middle Eastern Studies, who had come under fire for failing to represent Israeli perspectives.

But it has drawn the line at the White House's recent list of demands.