An aide to President Mahmoud Ahmadinejad told AFP that the attacks coincided with a planned visit by the president to the city, which is the capital of the oil-rich Khuzestan province.
However his visit was cancelled at the last minute due to bad weather.
Iranian television said the first bomb exploded outside a privately-run bank, killing six people and injuring 15.
The second blast, in front of a government office for natural resources, injured nine.
Ahvaz, which is dominated by ethnically minority Arabs, has been home to a wave of deadly unrest in the past year, including riots last April and a string of car bombings ahead of last June's presidential election.
The October bombings killed six people, and the June attacks killed at least eight.
Iranian authorities blame Arab extremists with links to foreign governments including Britain, a charge London has denied.