Hashemi-Rafsanjani was said to be leading as the vote count continued, Radio Farda reported, with some reports putting him at around 22 percent. There are conflicting reports over who is running second, but hard-line Tehran Mayor Mahmud Ahmadinejad and moderate cleric Hojatoleslam Mehdi Mahdavi-Karrubi were said to have between 17 and 20 percent each.
But not all the votes have been counted, and the standings are likely to change after the vote count in major cities, including the capital Tehran and Isfahan.
The Interior Ministry is expected to release final election results later today.
Going into yesterday's voting, Ahmadinejad and Mahdavi-Karrubi were widely regarded as outsiders.
Hashemi-Rafsanjani's main challengers were expected to be conservative candidate and former police chief Mohammad Baqer Qalibaf and the main reformist candidate, former Education Minister Mustafa Moin. But they were said to be in fourth and fifth place.
Reports that emerged while polling stations were still open yesterday suggested that Hashemi-Rafsanjani might be bound for a runoff against Moin.
The former hard-line head of Iran's State Broadcasting, Ali Larijani, and Iran's reformist Vice President Mohsen Mehralizadeh appeared to be faring worst among the seven candidates to replace incumbent President Hojatoleslam Mohammad Khatami.
Deputy Interior Minister Mahmud Mirlohi told reporters this morning that it has been a completely unpredictable election and that no one can foresee the results.
So far, the most ambitious pre-election prediction that looks set to materialize is that there will be an unprecedented second-round runoff between the top two vote getters, which would be held on 24 June.
Mehrdad Darvish, a professor of sociology at the university of Stockholm, said that one of the reasons for the elections runoff is that none of the Islamic Republic's camps presented a single candidate, so the votes were divided among the candidates.
"The election runoff for the first time in the Islamic regime indicates that diversity in the establishment's ranks has come to such an extreme that they can't get the society's votes by just bringing one prominent personality, an authority," Darvish said. "I think [that] no matter what the final result will be, in the long run we will be witnessing increasing tensions inside the Islamic regime."
No reliable turnout figures are available, but initial figures suggest it was around 67 percent in some provinces. That would be higher than expected and likely please the many officials who had urged a strong turnout.
Outgoing President Khatami, who is barred from a third term by the Iranian Constitution, said yesterday in Tehran that the election will strengthen the foundation of an Islamic democracy in Iran.
"I hope that with the enthusiastic participation of all women, men, and youngsters and adolescents in the elections, the difficult road to religious democracy, which is the result of the revolution, will be made smoother and easier and the decision of the Iranian nation will be respected by all of us," Khatami said.
[For all our coverage of Iran's ninth-ever presidential election, see RFE/RL's dedicated Iran Votes 2005 webpage.]