BEIJING—Apartment rental prices are soaring across China, posing a new challenge to Chinese authorities and compounding the threat of sky-high housing prices to the economy.

Rental prices for apartments are accelerating by double digits in 30 of China’s biggest, most vibrant cities. In Beijing, rents are up as much as 21% from a year ago, while in the south-central megacity of Chengdu they have climbed more than 30%, according to the China Real Estate Association, an arm of the country’s housing ministry.

This...