It was the first sign of a breakthrough between Lebanon's feuding sides, after the Annapolis conference eased tensions between their main backers, the United States and Syria.
Suleiman is seen as a neutral figure in a country where nearly every politician is considered either in the pro- or anti-Syrian camp. He is also seen as strong enough to ensure neither side dominates the other.
Lebanon's constitution bars a sitting army commander from becoming president. But on Wednesday, the largest bloc in parliament -- the anti-Syrian Al-Moustaqbal Movement -- announced it has dropped its rejection of amending the constitution.
The opposition has not announced its stance on Suleiman's candidacy, though he is respected among its leadership.
There also remain deeply divisive questions over how to change the constitution.
A major card remains: Christian Opposition leader Gen. Michel Aoun who has pushed his own candidacy for the presidency.
It was unclear whether Aoun would go along with Suleiman.
An economic delegation that visited Aoun Wednesday afternoon quoted the former army commander as saying that he does not mind amending the constitution, if need be, as long as it provides for the president to be chosen by the votes of the people directly.
Resigned Energy Minister Mohammad Fneish called for a political settlement, saying any consensus candidate has to be approved by the party which represents a majority of Christians. Fneish told The Daily Star that this party is the Change and Reform Bloc headed by Aoun. "Aoun is the starting point to any consensus. The question here is whether this move is a maneuver on the part of the ruling faction to create a rift between us and (Aoun) or is it a serious offer by the ruling faction and a clear position?" Fneish asked. He said that if Suleiman is considered a serious candidate by March 14, then his election should be discussed with Aoun first and the other opposition parties. Fneish said Hizbullah does not object to Suleiman in principle. Change and Reform bloc MP Ibrahim Kanaan urged the majority to prepare a draft proposal for a constitutional amendment and express a serious desire to support Suleiman so that Aoun's bloc can express its position. "Is this just a maneuver or a serious offer? Our experience with (the majority) in the past has not been very encouraging, many times they made overtures and went back on them," Kanaan told The Daily Star. MP Mohammad Raad, leader of the Loyalty to the Resistance bloc, told Naharnet that a constitutional amendment is possible only if Prime Minister Fouad Saniora resigns from the "unconstitutional government."
While some opposition figures voiced reservation over the March 14 offer, other leaders within the Hizbullah-led March 8 alliance raised questions over how to change the constitution.