A source close to Hariri said on Thursday that the former Lebanese prime minister is expected to leave Saudi Arabia for France within 48 hours.
Bahaa Hariri accused Hezbollah of trying to "control Lebanon", expressing gratitude to Saudi Arabia for "decades of support" for national institutions in Lebanon, as reported by AP.
Iranian President Hassan Rouhani meanwhile ratcheted up the rhetoric Wednesday against Saudi Arabia, his country's main regional rival, saying the kingdom pressured Hariri to resign in a "rare" intervention in another country's affairs.
Hariri became Lebanon's prime minister past year in a power-sharing deal in which Aoun, a political ally of Hezbollah, became head of state.
"Unless Hezbollah disarms and becomes a (solely) political party, Lebanon will be held hostage by Hezbollah and by extension Iran", he said.
Macron's office said the French leader would meet Hariri at noon on Saturday, with his family joining shortly afterwards for lunch.
"After speaking with Saudi Crown Prince Mohammed bin Salman and the Lebanese Prime Minister Saad al-Hariri, the president has invited Saad al-Hariri and his family to France", the Elysée said in a statement.
Lebanese President Michel Aoun (C) and Jordan's King Abdullah II (unseen) review the honor guard during an official welcome ceremony at Marka airport in Amman on February 14, 2017.
Macron spoke to reporters on Friday at an European Union summit in Sweden and said that "the role of France is to talk to everyone".
"If Mr. Hariri speaks from France, I would consider that he speaks freely, but his resignation must be presented in Lebanon, and he will have to remain there until the formation of the new government", Aoun said later in a statement issued by his office.
"I wait for the return of President (of the council of ministers) Hariri to decide the next move regarding the government", Mr Aoun said in comments published on his official Twitter account. He has yet to return to Lebanon from Riyadh, which had always been considered his chief political patron.
Hariri - whose father Rafik was also prime minister and was killed in a auto bombing in 2005 - became head of a shaky compromise government including Hezbollah previous year.
Mr Bassil added that Mr Hariri's resignation had left the people of Lebanon feeling deeply "frustrated" because in their eyes "he is still their prime minister".
Hariri announced his resignation from Saudi Arabia almost two weeks ago, citing concerns over the meddling of Iran and its Lebanese ally, the Shiite militant group Hezbollah, in regional affairs.
He accused Iran of sowing "discord, devastation and destruction" in the region.
France has however sought to maintain a nuanced position in the region.
Saudi Arabia is locked in a feud with Iran over regional influence; both countries support different groups in Lebanon.
Iran-allied rebels in Yemen fired a ballistic missile that was intercepted outside the Saudi capital earlier this month.
Hezbollah accused the kingdom of seeking to sow chaos in Lebanon.