If confirmed, the order would mark a stunning turnaround by the country's most powerful figure and offer hope to opposition forces who have waged street clashes to protest the re-election of President Mahmoud Ahmadinejad.
CBS News correspondent Elizabeth Palmer says, however, that the reports of an investigation being ordered can not be confirmed, and may be premature.
What CBS News can confirm is that the Guardian Council has received pro-reform candidate Mir Hossein Mousavi's official complaint of fraud, and has said it will report back within 10 days.
Palmer and her crew in Tehran report a near-complete lock-down on the flow of information both inside Iran, and getting out of Iran. Access to e-mail, and other Internet services is sporadic, at best. BlackBerry's are useless, and cell phones work intermittently. These factors, combined with the Iranian regime's long tradition of secrecy and propaganda, make getting any factual information out of Iran very difficult.
During a meeting on Sunday, Khamenei reportedly told Mousavi that he should be careful to proceed with his fraud allegations "through the courts," in a "legal" manner.
According to the Associated Press, Iran's State television quoted Ayatollah Ali Khamenei directing the Guardian Council, a high-level clerical panel, to look into the charges by Mousavi, who has said he is the rightful winner of Friday's presidential election.
Mousavi wrote a letter appealing to the Guardian Council and met Sunday with Khamenei, who holds almost limitless power over Iranian affairs. Such an election probe by the 12-member council would be uncharted territory and it is not immediately clear how it would proceed or how long it would take.
Election results must be authorized by the council, composed of clerics closely allied with the unelected supreme leader. All three of Ahmadinejad's challengers in the election - Mousavi and two others - have made public allegations of fraud after results showed the president winning by a 2-to-1 margin.
Palmer reports that, by sending his letter directly to the council, Mousavi snubbed the supreme leader's authority.
"Issues must be pursued through a legal channel," state TV quoted Khamenei as saying. The supreme leader said he has "insisted that the Guardian Council carefully probe this letter."
An important distinction to make is that Khamenei has asked the council to probe the "letter," not the reported fraud it details.
The day after the election, Khamenei urged the nation to unite behind Ahmadinejad and called the result a "divine assessment."
The results touched off three days of clashes - the worst unrest in Tehran in a decade. Protesters set fires and battled anti-riot police, including a clash overnight at Tehran University after 3,000 students gathered to oppose the election results.