Mr Cameron previously said in 2009, while he was opposition leader, that he would “like my children to go through the state sector”.
He also said it was “crazy” to pay high fees for private schools.
A Downing Street spokesman declined to comment on the Mail on Sunday report.
Last year, the Prime Minister’s 12-year-old daughter Nancy secured a place at Grey Coat Hospital School near Downing Street.
The top state school, which is now a Church of England academy, was founded in 1698 and Mr Cameron said his daughter being accepted was a “weight off everybody’s mind”.
