The two top schools are at the centre of a race row after a philanthropist claimed they rejected scholarships for poor white boys.
Dulwich College and Winchester College turned down Professor Sir Bryan Thwaites' offer to leave the funds in his will because they feared breaching anti-discrimination laws, The Times said.
Sir Byran, 96, attended both schools on scholarships and wanted to help disadvantaged white British boys because they perform worse at school than nearly every other ethnic group.
Recent studies show they are less likely to attend university than their peers and they perform relatively poorly in exams.
The row comes after the rap star Stormzy created Cambridge University scholarships to aid black British students exclusively.