The co-CEOs are thought to be quitting in the wake of a series of pratfalls, including a $2.5 billion fine for attempted interest rate manipulation, according to the publication.
The anticipated announcement also follows Friday's launch of an investigation into money laundering of about $6 billion from 2011 to 2015 through Deutsche Bank's Russian clients.
The bank was one of many financial institutions implicated in the 2008 real estate bubble that gave way to a global financial crisis. Deutsche Bank reached a settlement with shareholders early last year.
It is unclear yet who will succeed Fitschen and Anshuman as Deutsche Bank's next chief executives.