Under the old "direct grant" system, the Government paid for places for poorer pupils at selective private grammar schools - such as the prestigious Manchester Grammar School. However, Lord Adonis said the new style of school would no longer be selective.
But they would still offer pupils whose education was paid for by the state the chance to be taught in a school that had always been run as a successful independent school.
Two schools have already declared their hands. Belvedere Girls' School in Liverpool, where the millionaire philanthropist Sir Peter Lampl has been subsidising places for poorer pupils who pass its entrance exam, will drop its selection test and become one of the Government's flagship academies; as will the fee-paying William Hulme's Grammar School in Manchester, founded in 1887.
Several other unnamed schools have now told officials at the Department for Education and Skills they would like to follow suit. "We will re-establish a modern version of the direct grant schools," said Lord Adonis.
The biggest sponsor of the £5bn academies programme - with 13 academies either already open or in the pipeline - is the United Learning Trust, a Christian charity which previously had been responsible for running a chain of independent schools.
Lord Adonis's new style of direct grant school is an attempt to bring the ethos of the independent sector right to the heart of state education. He reveals his plans on the day it also emerges that a second member of the Russell Group of universities - the elite 19 higher-education institutions in the UK - is to sponsor an academy.
Liverpool University was the first Russell Group institution to support the academy programme. Now Nottingham University is to open a new academy on the site of William Sharp School - a struggling state school in Bilborough, Nottingham, where only a quarter of pupils have achieved five top grade A* to C grade passes at GCSE.
The new academy will specialise in health - using the facilities of the university's prestigious department of medicine - while student volunteers will act as mentors to pupils in an attempt to persuade pupils to go on to university.
Lord Adonis said in his interview with The Independent: "I doubt if anybody from William Sharp has ever gone on to Nottingham University." Lord Adonis said he would like to see more sponsorship from universities and business foundations for academies - where sponsors can run the school for an outlay of £2m.
"In five to 10 years, I would like to see every secondary school having links with external partners, including further education, higher education and employers," he added. He said the links would help make staying on in full-time education "routine for everyone". The UK lags 27th out of 30 in a league table of participation rates in education at 17 among Western nations.
Lord Adonis added that the target set by Tony Blair to establish 200 academies was ahead of schedule. "We will have 80 academies up and running by next September.
"The other 120 I'll have commissioned and have in the pipeline by next summer. The debate will then be how far beyond the 200 we want to go," he said.