Cognita, which runs 38 schools in the UK, was said to be talking to independent schools that are unhappy about changes to the law that mean they will have to prove their charitable status.
The Charity Commission is likely to insist they offer more places to children from poor families and do more work with state schools to qualify for tax breaks worth pound;100 million.
However, schools that have long enjoyed the benefits of being a charity will find it extremely difficult to drop the status and run as a private business.
A long winding-up process is required which is started by trustees telling the Charity Commission they no longer want to continue.
All of the charity's assets have to serve the original charitable purposes. For a private school, this would mean selling off its buildings and land and using that money, along with any cash in its coffers, in line with its original charitable aim. This might include donating the money to another education charity or setting up a charitable fund.
Only then could the school be taken over and run as a private business by a company such as Cognita, whose chairman is Chris Woodhead, a former chief inspector of schools.