The move would widen access to some of Britain's most privileged and academically successful schools, said Tony Millard, the association's chairman and headmaster of the independent Giggleswick School in Yorkshire.
He said the Government should help make boarding schools more socially inclusive. The schools could top up the bursaries themselves, he suggested.
"The important philanthropic initiatives supporting independent education in the past 12 months have bypassed the boarding sector," Mr Millard said. Yet the lifestyle of modern families made boarding more relevant and necessary, he argued.
Speaking before the opening of the association's annual coference in Torquay, he said, those who could benefit from boarding included: "Gifted children, disabled children, children from families under stress, children from busy families or whose parents are in expatriate employment."
The challenge was to find a way of making boarding more widely available at a cost that would not unacceptably favour a minority of state-funded pupils.
He said schools would be willing to top up state payments to meet the extra cost of boarding.
The number of full boarders - average fees, pound;13,500 - continues to decline. In 15 years, the total number of full boarders dropped from almost 109,000 to about 70,000. But many larger boarding schools such as Eton and Winchester, continue to flourish.
Of the Boarding School Association's 500 members, 36 are maintained.