Professor Steven Schwartz, the vice-chancellor of Brunel university, said they should be able to "vary the weight" given to A-levels depending on the background of pupils, including the kind of school they attended.
He said: "We want admissions tutors to be more broad-minded when they offer places. For example, there was an entire group of students who read widely but did not do well in their A-levels because their school did not prepare them for the exam. Each student needs to be looked at individually."
He said this would not give state pupils an unfair advantage over their privately-educated peers: "We do not want to bias the system in favour of private schools or state schools. If you look at grammar schools in Buckingham, you cannot say that going to a state school puts you at a disadvantage."
Although the report said pupils should not automatically be treated "more or less favourably" because of their background or school, it tempered this by saying admissions tutors should bear in mind that "many applicants have responsibilities at home or at work" and this could adversely affect A-level results. State students tended to perform better than their privately-educated peers at university.
The report also recommends that universities use a standard entrance exam similar to the United States' Scholastic Aptitude Test (Sat), that tests aptitude, potential and relevant skills. It would put an end to the raft of different entrance exams universities such as Oxford, Cambridge and University College London operate.
Dr Anthony Seldon, the head at fee-paying school Brighton College, said:
"The Sat test is a step in the right direction, but I think allowing the admissions tutor too much discretion could result in all kinds of unfairness. How can you distinguish between someone who's scrimped and saved to send their children to a private school and a pupil who's going to a chi-chi comp like the London Oratory?"
Universities do not have to follow the report's recommendations but Professor Schwartz has said it should be a "best practice" guide.
James Graham-Brown, head at the Royal High school, an independent girls school in Bath, said: "We've seen guidelines in the NHS too, and how they suddenly became targets on which funding is based. Universities should be free to select how they wish. Admitting more state school students might be right for Bristol university, but not for Oxford or Kent, for instance."
David Hart, the general secretary of the National Association of Head Teachers, welcomed the report but added: "Universities must be protected from any abuse by students who seek to put forward false reasons which explain away exam under-performance."
Cambridge university said taking into account pupil background was already the "cornerstone" of its admissions process.
The report also called for better training for admissions staff and for students to be able to apply for university places based on their actual rather than predicted A-level grades. Professor Schwartz's report is due out later this year.