Earl Spencer has written to Metropolitan Police chief Cressida Dick after an inquiry found Bashir had been “deceitful” in the way he went about securing the interview where Diana famously said there were “three of us” in her marriage to Prince Charles.
The police have previously said they will assess the contents of the Dyson Report which was published by the BBC on Thursday.
Earlier on Saturday, former BBC director-general Lord Tony Hall quit as chairman of the National Gallery.
Lord Hall, who was heavily criticised in the Dyson report for his botched inquiry into how the interview was obtained, said his continued presence at the gallery would be “distraction”.
The report by former master of the rolls Lord Dyson found the internal BBC investigation had covered up the “deceitful behaviour” by Bashir.
His resignation comes after another former BBC executive involved in the 1996 internal investigation, Tim Suter, announced on Friday he was stepping down from his board role with media watchdog Ofcom.
Lord Hall, who in 1996 was director of BBC news and current affairs, led the investigation which exonerated Bashir, even though he had previously admitted lying about the fake documents he used in obtaining the interview.
He was director-general when Bashir was controversially rehired by the BBC as religious affairs correspondent in 2016 and later promoted to religion editor.