The newspaper, citing unnamed sources, said authorities in the German state of Hesse have asked prosecutors to establish whether to open a criminal invvestigation into Deutsche Bank.
The report comes two months after the bank said it would ask external lawyers to investigate the activities of its own corporate security department, the FT said.
Deutsche Bank has terminated the contracts of two people as it continues its own inquiry into possible unauthorised surveillance of board members and investors.
The newspaper cites three cases, including one involving detectives who undertook surveillance on a board member who was suspected of leaks in 2001.
The bank later apologised to the board member.
Some cases involve outside contractors hired by the bank, the FT said.
The bank has refused to comment until its investigation is completed.