These additional scholarships will help universities to attract the brightest and the best students
Sir Steve Smith says reforms allowing institutions to take on unlimited numbers of top students next year will create competition for the brightest.
Institutions risk losing funding if they attract fewer top pupils than now.
Ministers insist there will be tough new criteria for attracting A-grade students from lower income backgrounds.
Sir Steve, the outgoing president of Universities UK - the body representing UK universities - said the government had introduced a market for A-level students with the best grades.
He said they might be offered cut price deals or bursaries to try to secure their places.
"The complication for universities is if you don't recruit the same percentage of students with AAB or better than you had last year what happens is you lose the funding for those students," he said.
"That means those students become very attractive and thus institutions will do what they can to lower the cost of attending university in order to attract them."
'Bidding war'
Sir Steve, Exeter University's vice chancellor, said many middle-ranking universities might reconsider their plans to charge the maximum fees of 9,000.
Even those with low numbers of grade AAB students will get involved in the bidding war for those with top grades - because they will be seen as a kite mark of an institution's success, he added.