Neither Benny Gantz not Benjamin Netanyahu has been able to pull together 61 seats for a majority.
It was their first round of direct talks since president Reuven Rivlin on Wednesday tasked Gantz with trying to form a government.
Gantz's nomination marked the first time since 2008 that someone other than Netanyahu has been asked by Israel's president to build a ruling coalition.
The prime minister has given up trying to form a governing coalition.
Blue and White and Likud are currently tied on seats after the country's second election in a year.
Benjamin Netanyahu's fight for a fifth term is an existential struggle.
Israel's longest-serving prime minister could find himself at the end of his political career because voters see him as too divisive and corrupt.