Figures published by the University and College Union show more than 40 universities have reported cases.
Just four universities, Nottingham, Manchester, Northumbria and Newcastle, account for more than half of the figures. Jo Grady, general secretary of the union, said: “What we are seeing now is a predictable and preventable crisis that ministers and universities chose not to do more to constrain.
The University of Bristol told hundreds of students on Friday to self-isolate after 40 people tested positive for Covid-19 in the same hall of residence. All 300 students in the Courtrooms accommodation in the city centre have been told to remain in their flats.
More than 1,600 students at two Tyneside universities, and another 100 in Durham, have tested positive for coronavirus in the past seven days.
Newcastle University said 1,003 out of its 28,000 students received a positive test result in the first week of October, and Northumbria University said on Thursday that 619 students have done so in the past week. That latest figure for Northumbria followed it announcing on Friday that it had 770 cases in the past week.
A spokeswoman for the Department for Education said: “Universities are prepared for a local outbreak and we have worked with them to help draw up plans for measures in the event of positive cases on campus, or a rise in cases locally.
“We recognise this has been a really challenging time for students, and will continue to do everything we can to help universities, and will continue to do everything we can to help universities ensure they have all the additional help and practical support they need.”