The new president travelled to Egypt on Monday to meet Egyptian President Abdel Fattah al-Sisi.
"Hopes of the role that Egypt could play are high. An Egypt of moderation and openness... could launch an Arab rescue initiative based on a strategy to fight terrorism," Aoun said at a joint press conference.
He said Egypt could "work on finding political solutions for the crises in the Arab world and especially Syria".
The two sides "agreed on the need to stand together against the dangers of terrorism", Sisi said, adding that Egypt was ready "to support the capabilities of Lebanon's army and its various security bodies".
Aoun, a Maronite Christian, was to meet later the same day with the leader of Egypt's Coptic Church, Pope Tawadros II, and also hold talks with Grand Imam Ahmed al-Tayeb of Al-Azhar, the highest institution of Sunni Islam.
On Tuesday, the Lebanese president is scheduled to meet Ahmed Aboul Gheit, secretary general of the Cairo-based Arab League.
Aoun, who was elected with the support of the powerful Iranian-backed Shiite movement Hezbollah, visited Saudi Arabia last month on a mission to patch up relations with Riyadh.
A Lebanese official source said at the time that Saudi Arabia and Lebanon had agreed to hold talks on restoring a $3-billion military aid package that Riyadh froze last year.
Mainly Sunni Saudi Arabia, a fierce regional rival of Iran, froze the aid deal over what it said was Hezbollah's dominance in Lebanon.
Aoun's election ended a two-year deadlock between Iran and Saudi-backed blocs in the Lebanese parliament.