A deal to end the higher education pay dispute is "within reach", with further talks with lecturers' unions to be held next week, universities have said.
Students will be hoping they can graduate this summer
The Universities and Colleges Employers' Association (Ucea) said a solution "satisfactory to all parties" was possible within days.
But the Natfhe union called Ucea "irresponsible and incompetent" for not making a higher pay offer on Friday.
Unions want a 23% increase, but employers are offering 12.6%.
Ucea's announcement follows five hours of talks on Friday aimed at ending a boycott of exam marking by lecturers.
A spokesman said employers said "good progress" had been made on Friday.
Employers had offered to open full negotiations with all seven unions representing university academic and support staff early next week, he added.
Natfhe and its sister union the Association of University Teachers have refused to mark exams or coursework since March. The AUT is also boycotting exam-setting.
Unions representing support staff - Unison, Amicus, the TGWU and the GMB - were invited to next week's talks.
Roger Kline, head of the universities department at Natfhe, said: "We are astonished that the employers have failed to make a final offer despite no prior indication that they would fail to do so.
"This is irresponsible and incompetent and most importantly it prolongs the difficulties for students."
The delay would "harden attitudes and ensure that the industrial action continues", he added.
Disruption for students
AUT general secretary Sally Hunt said: ┐Obviously we would have liked to have been in a position to discuss a pay offer with our members today, but that has not happened.
"What clearly does need to happen now is that an offer members can discuss is put on the table."
According to a survey of 85 institutions by the Press Association, 39% have been affected by the action.
Ucea says its 12.6% offer - over three years - is "final", but unions argue it is "too little".
The Commons education select committee has urged the two sides to talk.
Thousands of students face not graduating this summer if the dispute continues, with several student unions voicing concern.
Discussions earlier this month between unions and employers ended in failure and acrimony.