Pressure is growing for urgent reform of immigration rules amid concern many EU students enrolled at Scottish universities face additional costs running to hundreds of pounds due to the Covid pandemic.

Campaigners have warned first-year students will not qualify for pre-settled status if they arrive after the end of the transition period on December 31 – even though they may have opted to study online or could not move to the UK because of Covid-19 restrictions.

This would mean they will need to apply for a student visa when coming here in 2021 to continue their studies, hitting them with application costs of £348 and £470 a year in health charges.

Mantas Gudelis, who started a biochemistry degree at Edinburgh University in September and is studying at home in Lithuania, previously told The Guardian: “The health surcharge over four years is £2,000 and for my family that is a lot of money... The system should allow us to come because this is not our fault.”

Now the Scottish Government has written to Kevin Foster, UK Minister for Future Borders and Immigration, to ask that the rules be amended so all EU students who enrolled at a Scottish higher education institution in the 2020-21 academic year, but were unable to travel due to Covid, are deemed to have met residency requirements when applying to the Settlement Scheme.

This would be the case even if they arrive after December 31.

The letter was written by former migration minister Ben Macpherson – announced as Minister for Rural Affairs and the Natural Environment in a recent reshuffle – and will be taken forward by Europe Minister Jenny Gilruth.

It states: “The pandemic has meant that many universities and colleges have provided students with the opportunity to study remotely at home. A significant number of EU students have taken up this option, likely as a consequence of electing not to travel or because they are not allowed to leave their country of origin.

“As a result, some students who were scheduled to arrive in the UK to commence their studies in 2020 will not arrive until 2021 and, therefore, do not comply with the EU Settlement Scheme because they will not be living here by the end of 2020.

“This means they will need to apply for a student visa... Students in this position will be faced with visa application costs of £348 and an annual health charge of £470.”

The letter continues: “I am aware of a number of cases of students at Scottish universities, but currently studying remotely, who have been advised by the Home Office that they will incur the above costs if they wish to come to the UK from 2021.

“I am therefore writing to ask you to amend the Immigration Rules. All EU students who enrolled at a Scottish higher education institute in the academic year 2020-21 but were unable to travel here due to Covid restrictions should be deemed to have met the residency requirements when applying to the EU Settlement Scheme, even if they arrive here after 31 December 2020.”

The letter has received backing from university representatives.

“We support the ask made by the Scottish Government,” said a spokesman for Universities Scotland.

“More costs are a further barrier for EU students to come and study in Scotland. We hope the UK Government will recognise the extraordinary circumstances that many EU students find themselves in and make an exception.”

NUS Scotland President Matt Crilly said: “We have already called for an extension to the eligibility criteria for the EU Settlement Scheme until the end of the next academic year. The UK Government must make this a reality – and urgently.”

A Home Office spokeswoman said: “We have been clear that students... must be resident in the UK by 31 December 2020 to have rights under the Citizens’ Rights Agreements.

“The Government has undertaken extensive engagement work to make sure EEA (European Economic Area) and Swiss students understand what action they need to take in order to secure their right to be in the UK, and we are proud that there have already been 4.2 million applications to the EU Settlement Scheme.”