Berri, a leading opposition figure, had earlier held talks at the chamber with the governing coalition leaders, who enjoy Western backing.
Neither side controls enough seats in parliament to secure a two-thirds quorum required for the election.
A deal to elect Suleiman would ease Lebanon's worst political crisis since the 1975-1990 civil war and fill a post left vacant since the term of Emile Lahoud expired on November 23.
Al Jazeera's Rula Amin in Beirut said that government and opposition were still unable to reach an agreement on how to change the constitution so Suleiman could be elected.
"The opposition says they don't want to elect a president without resolving all the differences on all the other contentious issues, they want a package deal, a whole deal," she said.
"But the pro-government camp says 'No, there's a power vacuum, it's dangerous, they have to come to the parliament, elect a president and then sort out all other issues.'"
Arab and Western states are concerned that a prolonged vacuum in the presidency will further destabilise the country.
Condoleezza Rice, the US secretary of state, called on Sunday for the election to take place without foreign interference.
"Lebanon's neighbours, particularly Syria, need to encourage its allies and tell its allies to stop putting forward excuses for not going forward," she said.
The rival camps had been at odds for weeks over who should be the new president before consensus emerged Suleiman, who was appointed army chief in 1998 when Syria controlled Lebanon.
Suleiman is known for his good relations with the Hezbollah, which is backed by Iran and Syria.
But the election has been held up by differences including the make-up of the new government and means to amend the constitution to allow a senior public servant to become president.
Suleiman had been the opposition's preferred consensus candidate for the presidency, which must be occupied by a Maronite Christian according to Lebanon's sectarian power-sharing system.
The governing coalition has also backed his candidacy, saying it wanted to avoid a prolonged presidential vacuum.