The promise of greater rapprochement with Egypt is part of a package of incentives and efforts by Morsi to lure Iran, Syria's staunchest regional ally, away from Damascus and find an end to the bloodshed.
The two were meeting in Cairo as part of a Morsi-sponsored Syria peace initiative dubbed the "Islamic Quartet," bringing together Turkey, Saudi Arabia and Egypt - all supporters of the Syrian rebellion - with Iran.
Presidential spokesman Yasser Ali said that Morsi told the Iranian minister, Ali Akbar Salehi, in their meeting in Cairo that as president he cannot ignore public sentiment in Egypt, which is against the Syrian regime "that uses harsh language and violence against people."
The Egyptian leader delivered a similar message in person to Iranian President Mahmoud Ahmadinejad in Tehran during a meeting last month. He warned that Iran must end its support for Assad in order to prevent any chance of Western intervention in Syria.
It was also the first time that an Egyptian leader had visited Iran in decades. While in Tehran, Morsi delivered a stinging rebuke of the Syrian regime during a speech at an international gathering of so-called nonaligned nations.
Tehran cut ties with Cairo following the 1979 Islamic Revolution. Morsi has since moved to amend relations with Iran following his win in Egypt's first free presidential elections over the summer. Egypt's 2011 uprising, which ushered in those elections, helped spark Syria's own upheaval.
It remains highly unlikely that Iran would abandon Syrian President Bashar Assad as long as there is a chance for him - or at least the core of his regime - to hang on. Iran counts on Syria as a strategic outlet to the Mediterranean and a conduit to its anti-Israeli proxy Hezbollah in Lebanon.
Recently, a top commander of Iran's powerful Revolutionary Guard said Tehran had sent advisers to Lebanon and Syria, the clearest indication to date of Iran's direct assistance to Damascus.
Egyptian officials close to the presidency say Morsi has offered a package of incentives for Tehran to back off of its support to Assad, including the restoration of full diplomatic ties and efforts toward reconciliation with wealthy Gulf nations - a significant diplomatic prize for the Islamic Republic, especially as it comes under mounting pressure over its disputed nuclear program.
Sentiment is strong among Egyptians against Assad, whose regime they hold primarily responsible for an 18-month conflict in which 23,000 people have been killed, according to activists.
In a separate meeting in Cairo, the Iranian foreign minister told Arab League Secretary-General Nabil Elaraby that "gradual change (in Syria) is better than sudden change," according to diplomats there.
A day earlier, after having dinner with Elaraby, the U.N. envoy to Syria Lakhdar Brahimi and other foreign ministers, Salehi told reporters that the "the solution in Syria should be a Syrian solution," not "imposed from the outside."