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The rivalry between Uber Technologies Inc. and GrabTaxi Holdings Pte. Ltd. expanded to a new front Friday, when Grab said it was teaming up with self-driving car company nuTonomy in Singapore.

Selected Grab customers in the city-state will be able to register to use the ride-hailing app to book a nuTonomy vehicle. The rides will be free, though restricted to the relatively uncongested one-north business district and adjacent neighborhoods.

Founded by two researchers from the Massachusetts Institute of Technology, nuTonomy began testing its self-driving taxi service in Singapore last month with government approval, beating Uber’s trials in Pittsburgh by days.

The public trial is still small-scale: NuTonomy said it plans to have a dozen self-driving cars available by the end of the year; Grab has more than 40,000 drivers in the city.

Like Uber’s test fleet, Grab and nuTonomy will provide a backup human driver behind the wheel and a support engineer in the front passenger seat. If a trip goes outside the one-north business district, the driver will take control for that portion, Grab said.

Grab and nuTonomy didn’t provide specifics on areas of expansion from the new trial area, but said they were working with Singapore’s transport authorities to focus on areas where traditional taxis are difficult to find.

NuTonomy also says it is exploring options for expansion outside of Singapore.

Grab said the partnership, initially valid for one year, would allow the companies to better study their customers’ experience with on-demand self-driving cars. It also said it expects the partnership to provide new ways to improve its routing technology and mapping services, an area where it is trying to distinguish itself from Uber.

—Jake Maxwell Watts in Singapore contributed to this article. – – WSJ