Rouhani and Ayatollah Akbar Hashemi Rafsanjani, himself a former president, held second and first place respectively in the capital for the 88-member committee of clerics that appoints the country's supreme leader, according to the news agency.
With 1.5 million ballot papers counted out of 3.9 million cast, Rafsanjani was in first place with 692,000 votes. Rouhani was just behind with 652,000.
The figures were published by the semi-official ISNA news agency, citing the interior ministry which is responsible for managing Friday's elections to the assembly and to parliament. The elections are seen as ones that may influence the choice of Iran's next Supreme Leader, to replace the 76-year-old Ayatollah Ali Khamenei.
Reports in December said Iran's top assembly of clerics has actively begun looking for Khamenei's successor, likely due to his failing health.
Details of Khamenei’s health are normally kept under wraps in Iran, but in September of 2014 the Supreme Leader underwent successful prostate cancer surgery. At the time, the surgery and the media attention it received prompted speculations in Iran that Khamenei's health is deteriorating.
In another significant element, one of three hardline conservative ayatollahs that the pro-Rouhani "List of Hope" had urged voters to avoid backing may lose their seat on the assembly, according to the initial election results.
Mohammad Yazdi and Mohammad Taghi Mesbah-Yazdi. All three have adopted positions hostile to reformists, noted AFP.
Final results are expected on Sunday.
