The most high-profile candidate to register thus far is former hard-line President Mahmoud Ahmadinejad.
Iran's President Hassan Rouhani has registered to run in the upcoming presidential elections in May on the fourth day of the registration period which ends on Saturday evening.
The disputed re-election of Mr Ahmadinejad in 2009 triggered the biggest protests in Iran since the Islamic Revolution in 1979. According to the Associated Press, election officials were "stunned" when Mr. Ahmadinejad appeared.
In 2012, he became a prosecutor in the Special Court of Clerics, charged with disciplining the clergy, and spent two years as Iran's nationwide prosecutor-general from 2013 to 2015.
During Ahmadinejad's administration, Iran faced heavy global sanctions for its nuclear program, which Western countries said aimed to develop weapons.
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Millions of people demanded a re-run, but the supreme leader insisted the result was valid and ordered a major crackdown on dissent that saw dozens of opposition supporters killed and thousands detained.
Internationally, Ahmadinejad is more known for repeatedly questioning the scale of the Holocaust, predicting Israel's demise and expanding Iran's contested nuclear program. The latter will choose the country's next supreme leader. Baghaei registered alongside Ahmadinejad on Wednesday.
At least 638 have registered at the Ministry of Interior in Tehran in the first three days of the process through Thursday.
"Despite all the efforts of previous governments, the situation of the country is such that people ask why is there so much unemployment?" he said, adding that he would announce detailed economic plans at a later date. The council normally does not approve dissidents or women for the formal candidate list.
Khamenei and his hardline allies have strongly criticized the slow pace of economic revival since the lifting of sanctions past year, part of the nuclear deal with six major powers whereby Tehran agreed to curb its nuclear program.