Prime Minister David Cameron 'considering' sending son to £18,000 a year school
The Mail on Sunday said the Prime Minister and his wife Samantha had discussed registering Elwen, nine, at Colet Court Preparatory School.
The school, in Barnes, south-west London, is a feeder for St Paul's School, which was once attended by Chancellor George Osborne.
It describes itself on its website as "an independent school offering an outstanding, all-round education for gifted boys aged seven to 18 years". Colet Court costs £6,000 per term and charges fees for registering and entrance exams.
As opposition leader in 2009, Mr Cameron said he "would like my children to go through the state sector".
''I think it's crazy that we should pay lots of money for private schools. We all pay our taxes. You should have really good state schools available for all," he said.
He spoke last year of his relief after daughter Nancy, 12, secured a place at a top state school near Downing Street.
He said the news that she had been accepted at the Grey Coat Hospital School, in Westminster, which was founded in 1698 and is now a Church of England academy, was "a weight off everybody's mind".
Downing Street declined to comment on the report.