The February 26 report by the Jaras site has been confirmed by Hassan Yusefi Eshkevari, a religious scholar close to the opposition Green Movement who spoke to RFE/RL's Radio Farda, and also an unnamed source interviewed by the BBC's Persian Service.
The alleged meeting is said to have taken place at the unknown location where Musavi, who has been under house arrest since February 2011, is held. The report says that the opposition leader and former prime minister reacted defiantly to Khamenei's request.
"First of all, I will respond to the supreme leader about what I heard under the condition that there won't be any listening devices or cameras, and no one attends the meeting except [Ali Khamenei] and me," Musavi was quoted as saying by Jaras.
"Second, I want to be given the opportunity to speak to the people on a live television broadcast."
It's not clear whether Mojtaba Khamenei acted on his own or at the behest of his father, who is said to have personally issued the order for the house arrest of opposition figures Musavi, his wife Zahra Rahnavard, and reformist cleric Mehdi Karrubi.
The opposition leaders were put under house arrest after their call last year for an opposition rally that attracted tens of thousands of protesters. Both men have accused the authorities of massive vote rigging in the 2009 presidential vote and human rights abuses.
Eshkevari speculated that the meeting between Mojtaba Khamenei and Musavi was initiated by Iran's supreme leader, who has the last say in all state matters in the Islamic republic.
"I personally believe that [Mojtaba Khamenei] followed his father's orders. If this is the case, then Khamenei's goal is to find a way out of the current situation because the resistance of Musavi, Karrubi, and [other] political prisoners has created serious problems for Khamenei," Eshkevari said.
"He probably didn't think from Day 1 that the issue would be so serious. He wants a way out but at the same time, he doesn't want to make any concessions. He wanted to reach an agreement with Musavi behind the scenes."
Last week, the Kalame website reported that Musavi had told his daughters, in a telephone conversation, that he was standing firm on his previous stances. "Nothing has changed," he said, according to the opposition website.
Karrubi has also said, in rare meetings with his family, that he refuses to back down on his positions. Karrubi was also quoted in December by his wife as saying that the authorities were trying to turn the March 2 parliamentary elections into a sham vote.
-- Golnaz Esfandiari