The foreign minister is the son-in-law of Lebanese President Michel Aoun, who has not yet accepted Hariri's resignation and is awaiting his return before taking any decision.
"Today we demand the return to the nation of our Prime Minister Saad Hariri," tweeted Bassil.
Hariri announced in a televised speech from Saudi Arabia on Saturday that he was stepping down, citing Iran's "grip" on Lebanon and threats to his life.
The shock announcement raised fears that Lebanon -- split into rival camps led by Hariri and the Iranian-backed movement Hezbollah -- could once again descend into violence.
It also raised questions about the fate of Hariri, who also holds Saudi nationality, as it coincided with the announcement in Saudi Arabia of an anti-corruption purge in which dozens of princes, ministers and businessmen have been rounded up.
Al-Akhbar newspaper, which is close to Hezbollah, has suggested that Hariri was being held "hostage" in Saudi Arabia.
"We have no information (concerning Hariri). We just want him to return home," Moustapha Allouch, a member of Hariri's Al-Moustaqbal (Future) party told AFP, commenting on that report.
The party itself issued a statement on Thursday saying Hariri's return was "necessary" and essential for "total respect of Lebanon's sovereignty".
Saudi Crown Prince Mohammed bin Salman, right, meets with Lebanese Prime Minister Saad Hariri in Riyadh, Saudi Arabia on October 30, 2017. Source: AAP
On Sunday, Hezbollah chief Hassan Nasrallah said Hariri announced his resignation from Riyadh because it had been imposed by Saudi authorities, and wondered what would happen to the prime minister.
"Is he at home? Will they let him return? These are legitimate concerns," he said, referring to the anti-graft sweep in which Saudi Arabia now says 201 people were held.
As questions about Hariri's fate mounted, the resigned premier's media office on Thursday said he had met at his "Riyadh" home with the French ambassador to Saudi Arabia Francois Gouyette.
Two days earlier he had met in Riyadh with the US charge d'affaires and with the British ambassador Simon Collins.
On Tuesday, official media in the United Arab Emirates said Hariri visited Abu Dhabi where he met Crown Prince Mohammed bin Zayed al-Nahyan.
