The bank's stock added 1.04 percent to trade at €16.53 ($18.78) by 1010 GMT, against a DAX index of leading German shares down 0.73 percent.
Deutsche plans to move "large parts of the trading and investment-banking assets it currently books in London" to Frankfurt over the coming 18 months, Bloomberg reported, citing anonymous sources.
The decision to move could develop based on the outcome of talks between Brussels and London on their future trading relationship once Britain leaves the EU in 2019.
Most of the bank's 9,000 employees in Britain - around one in 10 of its 98,000 worldwide - are based in London, the largest presence of any foreign bank in the UK.
"The news isn't especially surprising in terms of substance and direction, even if the scope and the type of activities involved is large," a Frankfurt financial source who asked to remain anonymous told AFP.
Deutsche board member Sylvie Matherat said in April that thousands of bank jobs could be moved to EU territory after Brexit.
With Britain at risk of losing the "passport" financial firms use to deal with clients in the rest of the European Union when it leaves, employees in direct contact with customers may need to be based on EU territory in future.
Other jobs will need to move to deal with business that must be booked in the European Union, as will risk management workers, who must be based in the EU to satisfy banking supervisors' requirements.
Deutsche Bank refused to comment when contacted by AFP.
Several other banks, especially Asian institutions, have recently announced that they would move European headquarters from London to Frankfurt.
In another possible driver of gains for Deutsche stock Thursday, chief executive John Cryan told German weekly Die Zeit that the bank expects to return to profit this year.