Ministers face renewed pressure for changes to key elements of the higher education bill despite their dramatic Commons victory this week.
Rebel Labour MPs were confident of overturning the five-vote margin to remove variable fees at a later stage of debate. And universities are pre-paring a campaign focusing on the remit for the Office for Fair Access.
Vice-chancellors have kept major concerns about the bill under wraps while waging a desperate battle to get the legislation past its second reading, a document leaked to The THES reveals.
A discussion document produced by Universities UK officials immediately after the publication of the bill earlier this month indicates serious concerns about the powers given to the education secretary, particularly in relation to Offa.
UUK is also fearful that Offa will leave universities open to litigation from disappointed students and that it could end up dictating university admissions and the social composition of the student body.
Concerns about Offa are shared by many vice-chancellors. In an opening salvo in what some predict will be an "almighty battle", Sir Richard Sykes, rector of Imperial College London, said this week that if Offa was too onerous, he would forgo fee money and recruit overseas and postgraduate students instead.
The document also reveals UUK's determination to ensure that fee money is not eroded by cuts in public funding. It proposes that any attempt by the government to determine how income from fees is spent should be subject to a vote of both houses.
In an official statement this week, Ivor Crewe, UUK president, says: "This is a landmark bill that represents a critical point for the future of higher education."
But the closeness of Tuesday's vote will ensure that ministers face further battles in committee and in the House of Lords. Some Labour rebels had last-minute changes of heart and may still support amendments seeking to outlaw variable fees.
Rebel Labour MP Nick Brown, whose last-minute conversion saved the bill, said: "We have strong medium-term safeguards to ensure against any further moves towards the marketisation of higher education."
The UUK document raises questions about whether UUK should push for the graduate contribution scheme to be extended to part-time students, who are currently subject to upfront fees. Such a move would be expensive and politically sensitive. The Open University did not sign an advertisement taken out by UUK in Monday's Guardian that called on MPs to support the second reading of the bill.
The paper was due to be discussed by the UUK board this Friday. Only after this meeting will a UUK strategy emerge.
Concerns about Offa make up the majority of the document. It welcomes the government's decision to scrap the requirement that universities put aside a third of fee income for bursaries but stresses the need to remain alert about the extent to which the secretary of state intends to "direct, through Offa, institutional strategies for widening participation and, in particular, the provision of bursaries".
The bill's statement that access agreements must be in relation to each qualifying course is described as "wholly impractical".
The document says: "There are about 50,000 courses for which such agreements would need to be reached, and any system that attempts to treat them on an individual basis must be extremely bureaucratic and burdensome."
UUK proposes that access agreements be in relation to "general fee policy". It wants scope for subjectivity on Offa's part removed from the bill and clearer definitions of "underrepresented groups".
Crucially, UUK fears that the wording of the bill opens the way for Offa to move away from monitoring efforts to widen participation to monitoring the composition of the student body.
It also proposes amendments to ensure that students cannot take universities to court on the basis of perceived failures in access plans.
It also wants universities to be able to modify their five-year access plans within that period.
The Coalition of Modern Universities, which recently rebranded itself Campaigning for Mainstream Universities to emphasise that it represents the interests of the vast majority of universities, also published a statement.
"We will continue to work with all the parties involved to ensure that both universities and students receive a package of measures that will succeed in encouraging greater participation in higher education," it says.
The CMU last week published a six-point agenda on key changes it wants as the bill passes through parliament. A key issue is a floor on the unit of funding.
Michael Sterling, chair of the Russell Group and vice-chancellor of Birmingham University, regretted the bill had passed with such a small majority and said he was fearful of further concessions. He said he was hoping for amendments in the House of Lords to reduce the intrusive nature of Offa.