Aspinall, 32, has been the interim champion for 18 months and has publicly said he believes Jones is retired.
White has been insisting since last November, when Jones last fought, the bout with Aspinall would happen next and clarified that - while Jones had agreed to the fight - he had not signed a contract.
"John agreed to fight Tom," he said.
"Jon Jones agreed to do the fight. Jon Jones was offered the deal and accepted the deal."