Citing unnamed sources, the paper said that the deal to take over the UK's fourth-largest wealth manager could be worth as much as Â£350 million.
Deutsche Bank is apparently now in "deep discussions" with Tilney's parent company Bridgepoint, a private equity firm, about the purchase.
Tilney is estimated to have around Â£6.7 billion in assets under management, servicing more than 15,000 clients following several new strategic takeovers under the leadership of its chief executive David Campbell.
The Financial Times reports that Tilney last year bought Clydesdale Bank following the acquisition of the private client divisions of Edinburgh Fund Managers and SG Hambros.
There is expected to be a growing demand for private client wealth management services in the coming years, due to the increasing number of people worldwide with large amounts of money to invest.