The report said that in both cities the bombs were planted in the office of the local governor.
Dezful and Abadan lie in the oil-producing province of Khuzestan, which has seen ethnic tensions between Iranian authorities and the Arab minority over the past year.
The provincial capital Ahvaz, home to a large community of ethnic Arabs, has been hit by a string of bombings in recent months.
Iranian officials have accused British forces in neighboring Iraq of being behind the unrest. Britain denies the charges.
(Reuters, AFP)