"The only thing banned here is failure or a reaching a dead-end."
Al Jazeera's Sherine Tadros, reporting from Baabda in west Lebanon, said: "[It] is the beginning of a discussion on how Hezbollah and the Lebanese state can co-exist."
Tadros said: "It is a much wider discussion than simply how to disarm Hezbollah. Hezbollah have already said that it will not disarm as long as there is a threat to the Lebanese state."
Despite mixed feelings towards the disarming of the Hezbollah, Sleiman said in a televised address on Tuesday that Israel was the "great danger" that threatened the country.
"We are all agreed that Israel is the source of great danger to us," the president said.
"It does not refrain from declaring its agressive intentions aginst us and it thretens our national institutions and facilites.
"I am comletely confident we can adopt a strategy that protects Lebanon."
A statement from Sleiman's office, released after the three-hour meeting, said the next session would be held on November 5, adding that the president would mediate bilateral talks in the meantime to "consolidate the reconciliation".
United in defence
Hezbollah, considered a terrorist group by the United States, has refused to disarm arguing that its weapons and guerrilla army will help defend the country against neighbouring Israel.
But members of the Western-backed majority in parliament argue that Hezbollah's weapons undermine the authority of the state, which should be the sole decision-maker on matters of defence.
Hezbollah seized control of west Beirut in May in clashes that left 65 people dead [AFP]
Miguel Angel Moratinos, Spain's foreign minister, said during a visit to Lebanon that Hezbollah's weapons were an internal issue "to be discussed on the negotiating table".
Al Jazeera's Tadros said: "A Hezbollah official in the south said on Monday night that they were ready to begin discussions on how there could be a complimentary relationship between Hezbollah fighters and the Lebanese army."
Tadros said that Walid Jumblatt, a prominent member of the March 14 group, said on Monday that he was ready for the state to utilise Hezbollah's military power.
"That is a very significant change in the language that we have seen in the past," she said.
Hezbollah's arsenal became an even more sensitive topic after it staged a spectacular takeover of mainly Sunni areas of west Beirut in May in an explosion of sectarian violence that left 65 people dead and raised fears of a return to the days of the 1975-1990 civil war.
Demands for the disarmament of Iran-backed Hezbollah, a group with broad support among Lebanese Shia, have been at the heart of more than three years of turbulence in Lebanon following the 2005 assassination of Rafiq al-Hariri, a former prime minister.
An editorial in Lebanon's An-Nahar newspaper, which is close to the anti-Syrian ruling coalition in parliament, said: "All Lebanese are hoping that the dialogue will be successful, but a real solution will only be found when Hezbollah stops using its weapons to serve regional interests and stops imposing its hegemony by force on the local scene."
The talks are an outcome of the Doha peace accord signed in May.
The accord curtailed 18 months of deadlock, during which the sides were unable to agree on a new president.
Sleiman was eventually appointed and the formation of a national unity government went ahead.
But the country has seen sporadic violence since.
On Tuesday morning, one person was shot dead and another two wounded in the split Sunni and Shia village of Taalbaya Bekaa Valley region in eastern Lebanon.
In June, three people were killed in the same region in clashes between supporters of Hezbollah and the Future Movement, a rival faction.
Six makeshift bombs exploded early on Monday in a mixed Sunni-Shia area of west Beirut and two similar bombs were defused by the Lebanese army near a church.