The Paris-based media-rights group says Adnan Hasanpur and Abdulvahed Hiwa Butimar have been on hunger strike for nearly 30 days.
They were sentenced to death after being found guilty of acting against Iran's national security.
"Their most basic rights were violated as they were barred from court when the sentence was handed down,” according to the letter from RSF. “Even more egregiously, they were not notified of the sentence and only found out from a newspaper.”
The RSF says it has asked Ban to urge Iran to adhere to the international treaties it has signed concerning civil and political rights.
The group says 11 journalists are currently being held in Iranian prisons.