Beijing and Shanghai have given no indication of whether the bans will be lifted.
Chinese bike-rental firms, Mobike and Hello Transtech, expressed their willingness to enter the bidding.
The rules went into effect on Wednesday on Didi’s bike-rental platforms Qingju and Bluegogo.
Didi’s struggles in the bike-rental market led to a launch of its own brand, Qingju.
Cash-strapped Ofo could try wheeling its business down another path.