The crisis is posing the biggest threat to Lebanon's stability since the 1975-1990 civil war, and a new cabinet could help initiate reforms to unlock foreign aid.
"Government formation is the only alternative to the chaos that threatens everyone," Hezbollah's parliamentary bloc said in a statement, calling for transparency and cooperation between Lebanon's leaders.
The row between Hariri and Aoun intensified after a meeting between the two to discuss a new cabinet ended in a public repudiation.
Lebanon's Iran-backed Hezbollah group is an ally of Aoun.
Its chief Hassan Nasrallah has said he would support a cabinet of specialists should Hariri and Aoun come to an agreement.
Diab said on Wednesday it was up to parliament whether to re-activate a resigned cabinet.