The app is expected to eventually expand to include other features like transit schedules, real-time traffic information and bike sharing information.
The measure is expected to go before the Houston City Council next week for final approval.
Houston, the nation's fourth largest city, is looking at improving transport options as plans progress for hosting the Super Bowl in February.
Officials say no taxpayer money would be used to develop or advertise the app, but city staff would guide the process with Arro. Elected officials would promote it.
Tina Paez, director of Houston's Administration and Regulatory Affairs Department, said "We're excited to bring the taxi industry, fully, into the digital age."
Arro already handles taxi dispatch in several cities, including New York, Chicago and Boston.
The taxi and limo industry faces challenge from ride-hailing services such as Uber. The industry encouraged the city to develop something to help even the playing field with Uber, The Houston Chronicle reports. An Uber spokeswoman had no immediate comment Monday.
Houston Transportation and Lone Star Cab are the city's two largest companies, and hold about three-quarters of the taxi permits in the city. Cindy Clifford, spokeswoman for Houston Transportation, said the company supported the new app, despite already having its own systems.
"We are for any way to make it convenient for someone to get a taxi," Clifford said.