Mikati, appointed this week after the Hariri's pro-Western government was toppled, did not make any comments after his meeting with President Michel Sleiman.
The telecoms tycoon and former premier held two days of talks with parliamentary groups on forming his government, ending on Friday with the Hariri camp still refusing to join his administration.
A source close to Mikati said that Mikati's cabinet would likely consist of 24 to 30 ministers, most of whom would represent political parties but which would also include technocrats.
"Contacts are ongoing with all parties in order to secure the widest participation possible in the new government," the source said.
"Mr Mikati wants to form his government quickly but not hastily."
The Hariri government collapsed on January 12 when Hezbollah and its allies pulled 11 ministers from cabinet in a dispute over the Special Tribunal for Lebanon (STL).
The Netherlands-based STL is investigating the 2005 assassination of Rafiq Hariri, father of the outgoing premier, and is expected to implicate Hezbollah operatives in the murder.
The Shiite group has warned against any such accusation and is demanding Lebanon end all cooperation with the tribunal.
A Hariri-led alliance in turn is demanding Mikati, a political moderate with good ties to former powerbroker Syria, guarantee he will not disavow the tribunal.
Saudi-backed Hariri has refused to join a cabinet led by a prime minister whom he says was appointed by the Shiite Hezbollah.
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