The investigation was taken over by Sue Gray, a former head of propriety and ethics at the Cabinet Office and a senior civil servant in the Levelling Up department.
She handed her finding to the Met, which launched a criminal investigation that is ongoing.
On Friday night the force said it would not comment on individuals, while the Cabinet Office said it would be “inappropriate to comment on speculation”.
The Financial Times reported concerns from current ministers about whether Mr Case could continue in his role as head of the Civil Service if he is found to have broken the law, or if his own staff had broken the law with his knowledge.
The Met is expected to conclude its investigation before Dame Cressida Dick, its current commissioner, stands down in mid-April.