According to Bloomberg Businessweek, Uber’s chief executive Travis Kalanick said the launch would take place in Pittsburgh, Pennsylvania.
At first, the vehicles will be supervised by a driver, who can take control if necessary, and an observer.
Uber eventually hopes to replace its one million drivers, Bloomberg said.
The firm has not yet responded to a request for comment.
Customers in Pittsburgh will request rides as usual via the Uber app but trips in self-driving cars – allocated at random – will initially be free, according to Bloomberg.
There were conflicting reports over which vehicles would be involved in the first trials.
Some of the first self-driving Uber cars in Pittsburgh will be modified XC90s, reported Bloomberg.
But the Associated Press said the initial cars would be modified Ford Fusion.
Volvo has confirmed it is taking part in a “joint project” between Uber and the carmaker.
“Both Uber and Volvo will use the same base vehicle for the next stage of their own autonomous car strategies,” the firm said.
“This will involve Uber adding its own self-developed autonomous driving systems to the Volvo base vehicle.”
Engineers from both companies would collaborate on the project, the firm added.
“This alliance places Volvo at the heart of the current technological revolution in the automotive industry,” said Hakan Samuelsson, president and chief executive of Volvo.
The Bloomberg story notes that the Volvo deal will not be an exclusive one and that Uber plans to work with other carmakers.
Uber’s hard-charging founder Travis Kalanick has been outlining a vision of a self-driving fleet for some years – but it always sounded like a long-term vision akin to Elon Musk’s plan for a settlement on Mars.
Indeed, two years ago the company’s operations director Ryan Graves told me that focussing on self-driving cars was a “fool’s errand” in the short term and that nobody would be getting in one in the next few years.
So the news that self-driving Ubers will be available to customers in Pittsburgh later this month comes as a huge surprise. It will be a wake-up call to Google – which has done so much work in this field but has yet to launch any kind of commercial service – and to Mr Musk’s Tesla, currently battling adverse publicity over the reliability of its Autopilot function.
It is also a measure of just how much money Uber has invested in its automated driving research programme – but having raised over $10 billion (£7.6bn) from backers, the company has plenty of financial firepower at its disposal.
One caveat however – the “self-driving” Volvos picking up Uber users will still have a human in the driving seat as a supervisor. Even if Uber was confident enough in the technology to dispense with a human, neither Pittsburgh nor any other city is ready to adapt its regulations to allow completely autonomous driving.
And that means that Uber’s latest coup is an impressive piece of marketing for its vision of the future of transport, but will prove a very expensive way of ferrying people around for some years to come.
Some of Uber’s know-how in the field of self-driving vehicles has come via its agreement to purchase Otto, a start-up founded by former Google employees to develop driverless trucks.
Otto was co-founded by Anthony Levandowski, whose previous start-up, 510 Systems, was bought by Google when it began its own self-driving vehicles project.
Volvo has already been testing self-driving versions of the same vehicle in Sweden as part of its “Drive Me” project, said Alan Stevens at the UK’s Transport Research Laboratory (TRL) and the Institution of Engineering and Technology (IET).
The firm has also trialled the vehicle in Australia and plans to do so in London next year.
“I think it works well and I think it’s a good small step to be taking,” Mr Stevens told the BBC,