that the number of children from low-income families receiving fee assistance was likely to be higher than it appeared, as some scholarships were awarded to pupils whose families could not otherwise afford fees.
But he called on private schools to use means testing to make sure support was given to those who needed it most.
"The higher the proportion that can be means-tested, the better, because it's a more accurate way of targeting money at the issue most of us want to approach, which is making our schools affordable," he said. "Over the past 20 years, schools have reduced the amount of money that is given in scholarships to pupils who don't need them."
Funding for "eligible families" should not be scaled back, Mr Lenon said, because many of these were "very deserving" and some of the money had been donated to schools precisely to support these pupils.
But Lee Elliot Major, chief executive of the Sutton Trust, which aims to promote social mobility, said "much more" needed to be done to open independent schools up to pupils "on the basis of ability rather than ability to pay".
The charity said schools should "look at maximising the extent to which bursaries are targeted at those with modest means" by making greater use of means testing. It also published a report last year which argues that the system needs to be "transformed" because "today, unless your parents can find pound;12,500 a year after tax, access is by and large denied".
Adam Pettitt, headmaster of the independent Highgate School in London, said many schools had moved away from scholarships offering fee remission in recent years.
The ISC census shows that 8 per cent of private-school pupils received means-tested bursaries in 2014. In addition, 12.8 per cent received help under an "eligible families" programme and 10.2 per cent received scholarship funding.
Since 2009, when the figures were first recorded, the total value of means-tested bursaries has risen by a third. This, however, has been outstripped by a 43.6 per cent increase in "eligible families" funding.
Schools spent pound;364 million on non-means-tested bursaries last year, the census shows, compared with pound;341 million on means-tested support. However the average amount spent on means-tested pupils was pound;8,227 - far more than the average of about pound;3,000 given to pupils from "eligible families" and those receiving scholarships.
The findings come after Eton College headmaster Tony Little last week said that private boarding schools should consider offering bursaries to families earning as much as pound;80,000 per year. In a speech to the Boarding Schools' Association's annual conference in London, Mr Little said middle-income families were being "squeezed out" of boarding school education because of rising fees.
Senior policemen, GPs and airline pilots had traditionally sent their children to boarding schools, he said, but could no longer afford to do so in many cases.
`Our reach was limited'
Highgate School in North London stopped offering fee assistance through scholarships three years ago. This was because headmaster Adam Pettitt took the view that "subsidising the cost of education without having a view of whether this was necessary or not limited our ability to reach people who wouldn't otherwise be able to come to the school", he says.
Highgate has also stopped offering special discounts to families of clergy members, although some of these children still get places at the school through its means-tested bursaries. "The governors decided that because those people qualified anyway, why should we stipulate?" Mr Pettitt says.
He adds that his school targets its bursary funding at those with low incomes.