The wage dispute comes as Germany's largest lender tries to keep a lid on costs and manage customer service problems at Postbank.
The union has demanded a 15.5% pay increase for the thousands of workers affected and has held a series of strikes in the standoff.
Deutsche Bank raised its offer to a 10% pay increase in two steps during Tuesday's negotiations, one of the people said.
Deutsche Bank declined to comment. A Verdi official said the union wasn't able to immediately comment.
The union made its initial demand for a 15.5% pay increase in December, a level it says is justified due to the rapid pace of inflation after a previous wage agreement was concluded in 2022.
(Reporting by Tom SimsEditing by Miranda Murray and Louise Heavens)
By Tom Sims