The explosion targeting the bus occurred on Shahrah-e-Quaideen, a key thoroughfare in the city, at 3.05 PM local time.
Provincial Health Minister Sagheer Ahmed said 12 people were killed and 50 injured in the attack.
There was confusion over the nature of the blast.
TV news channels quoted witnesses as saying that a motorcycle-borne suicide bomber rammed the bus.
However, Karachi police chief Waseem Ahmed said initial investigations had indicated that an explosive device planted in the motorcycle was triggered by remote control as the bus was passing by.
The second blast occurred around 5 PM local time outside the emergency ward of Jinnah Post Graduate Medical Centre hospital, where most of the dead and injured of the first attack were taken.
Geo news channel quoted officials and witnesses as saying that 10 people were killed and several others injured in the second blast. Several ambulances were also damaged in the attack.