The data recorders, similar to those installed on commercial airplanes, would track the taxi's movements and speed.
The move is part of Mayor Bill de Blasio's Vision Zero plan to eliminate traffic fatalities.
A bill to start a pilot program is expected to be introduced in the City Council Thursday.
Reports say the black box data would not be used to penalize drivers during the test program, but that could change if the program becomes permanent.
The city could use the information to fine drivers, or they could develop technology that to stop the meter if a driver exceeds a certain speed.