The 12-member Guardian Council said in May eight of the more than 500 registered candidates qualified to have their names appear on Friday's ballot.
Adel, the first non-cleric to host the position of speaker, said Monday he was pulling out of the race, but would stand in solidarity with Principlist candidates, who are considered conservative lawmakers.
"I stay on my word and declare that I quit the Islamic republic of Iran's presidential election race to contribute my share to the victory of the Principlists in this election," he was quoted by the semiofficial Fars News Agency as saying.
Adel is considered a close ally of Iranian Supreme Leader Ali Khamenei. Rival candidates Saeed Jalili, Iran's chief nuclear negotiator, and former foreign policy adviser Ali Akbar Velayati are also close to the ruling cleric.
The Guardian Council is permitted to re-examine the candidates before the election. Council spokesman Abbas Ali Kadkhodaei denied suggestions the clerical body was vetting reformist lawmakers before Election Day.
Candidates wrapped up their last debate Friday on foreign policy, vowing to maintain a functioning nuclear research program.