The offender "had sprayed the tombstone with white paint," ISNA news agency quoted Ahmadi Kamali, the governor of central city of Ardekan where the late Khatami is buried, as saying.
Kamali said the paint covered an inscription on the grave bearing a message from the late Ayatollah Ruhollah Khomeini, who spearheaded the Islamic revolution of 1979.
"We condemn this action and an individual has been arrested in connection with this," he added.
Mehr news agency said the tomb was vandalised on Sunday night.
Ayatollah Khatami was the representative of Khomeini in Yazd province.
His son Khatami, a reformist, has himself come under fire from hardliners since he became a major supporter of the opposition movement which refuses to acknowledge last year's re-election of President Mahmoud Ahmadinejad.
He was even reportedly attacked by hardliners on February 11 while on way to participate in the traditional rally marking the 31st anniversary of the Islamic revolution.
