Mitchell is head coach at the Blue Bulls in Pretoria and was assistant to Clive Woodward with England between 1997 and 2000.
The 54-year-old’s contract in South Africa runs until 2019, so an early release would need negotiating.
The Rugby Football Union (RFU) refused to comment when approached by the BBC.
England have been without a defence coach since Gustard’s departure to Harlequins following the tour of South Africa in June.
While the Bulls have stressed Mitchell is under contract, it is believed the RFU would be prepared to pay compensation.
The RFU itself received a financial package from Quins to secure Gustard’s early release from his England deal, which was due to expire after the 2019 Rugby World Cup.
Mitchell’s prospective appointment would be a bold one by Jones.