Beijing and Shanghai have given no indication of whether the bans will be lifted.
Other bike-rental companies are considering raising prices, as warmer weather brings a likely rebound in trip numbers.
Chinese bike-rental firms, Mobike and Hello Transtech, expressed their willingness to enter the bidding.
The law comes as concern mounts about the financial stability of a number of mobility firms in the rental economy sector.
Foshan will require Mobike, ofo, and other bike rental companies to hand over users blacklisted for bad parking.