Rafsanjani shunned the inauguration ceremony of Ahmadinejad on August 5, as well as Khamenei's endorsement ceremony amid complaints that Ahmadinejad's re-election was fraudulent.
"We demand a legal probe on the basis of Article 111 of the Constitution, which is a responsibility of the Experts Assembly," stated the letter, which the Journal said was written by the head of the organisation of former reformist parliamentarians.
Article 111 says if the supreme leader "becomes incapable of fulfilling his constitutional duties", he will be dismissed, the paper said.
The letter marks the first time a political group has questioned the authority of the supreme leader, the paper reported.
The complaint comes two months after the June 12 election which returned Ahmadinejad to power but which the opposition says was rigged.
Mir Hossein Mousavi, a reformist presidential candidate in the elections, and Mehbi Karroubi, another opposition candidate, have led protests in the wake of the elections, sparking a major crackdown by the police.
Mousavi, many of whose supporters remain in detention, has been accused by Iran's Basij militia of several crimes, including undermining national security and it has demanded an investigation into his conduct.
Not above the law
The Journal said Khamenei is not above the law, citing Iran's constitution, adding that the assembly has the right to review his performance as a religious and political leader.
Saying the letter is not likely to result in any action from the assembly as the hard-liners have marginalised the reformists, the paper said the move signals an important turning point for the reform movement.
Criticising the supreme leader was once unthinkable, and had serious repercussions, according to The Journal.
But demonstrators now regularly chant slurs against Khamenei and write insults about him in green spray paint on walls in the capital, said the paper.