Chief inspector Chris Woodhead ordered a further inspection of primary teacher training last year after he decided that inspectors had painted too rosy a picture of teacher training. A more stringent framework was subsequently put into place.
The institutions at risk are Durham University, Bath College of higher education, Warwick University, and Derby University. Warwick has taken legal advice.
The stumbling block for all four has been their ability to assess students' performance. The Office for Standards in Education inspectors did not agree with the universities' evaluation of their students and judged it to be inconsistent.
Almost all primary teacher-training courses have now been reinspected. When the process is complete the Teacher Training Agency will have to decide if funding should be withdrawn.
Mr Woodhead had said the follow-up survey would focus more on English and maths, but the decision provoked widespread suspicion in the sector.
The London Institute of Education was so anxious about the reinspection motives that it fought for a delay and will not be inspected again until next year.
Under the framework devised by OFSTED and the Teacher Training Agency, courses are judged across a series of areas divided into "cells". A bottom grade (4) in just one cell means that the university's entire teacher-training programme is called into question and deemed not to comply with the Government's criteria. The TTA can then consider withdrawing accreditation and funding.
A Durham University spokesman confirmed that its primary course was given a grade 4 on just one cell - the way it and its partner schools assessed students' performance.
"We are surprised and obviously very concerned about the implications for funding. We were inspected while our courses were in a period of transition, and feel inspectors have not taken this into account. Vigorous discussions with OFSTED are continuing." Durham received a grade 1 (outstanding or very good) and three grade 2s the first time around.
Warwick has also been told it is failing to judge students' ability to teach number and reading adequately and has been given grade 4s in two cells. It has taken legal advice but decided to negotiate through OFSTED's complaints procedure. Warwick's overall gradings last time were three "goods" and one "sound".
Derby, judged sound overall last time, confirmed it had been failed on two cells relating to the university's assessment of students.
Bath College of HE (three goods and one very good first time) is believed to have been failed on the same basis.
The Committee of Vice Chancellors and Principals has called a meeting with OFSTED today to discuss the universities' position. "Some of them are extremely concerned about their position and the conduct of the inspection," said policy adviser Patricia Ambrose.
Mary Russell, secretary of the Universities Council for the Education of Teachers, said it was absurd that "the future of a whole teacher-training programme could depend on a difference of opinion between HMI and a university over one or two students."
Peter Mortimore, director of London University's Institute of Education, said he was "extremely alarmed" that courses had been failed "because of differences in grades given by the course tutorpartner school and the OFSTED inspector or associate inspector to one or more students".
He said it was unrealistic to expect agreement in the grades awarded to borderline passfail students. "It flies against everything we know about psychometrics and the measurement of skills".