After four inconclusive parliamentary elections in two years, a 28-day mandate for opposition leader Yair Lapid to form a new government runs out on Wednesday, and media reports said he was close to putting together a coalition that would end Netanyahu's 12-year stretch as prime minister.
Lapid's chances of success rest largely with far-right politician Naftali Bennett, a kingmaker whose Yamina party has six key seats in parliament.
Bennett, 49, was widely expected to announce, possibly as early as Sunday, whether he would team up with Lapid, who leads the Yesh Atid party.
But first, Bennett would have to rally his own party's legislators behind joining what Netanyahu's opponents have described as a government of "change" comprising factions from the left, centre and right.
Still short of a parliamentary majority after a March 23 election that ended in stalemate, such a diverse grouping could be fragile, and would require outside backing by Arab members of parliament whose political views differ sharply from Yamina's.
Bennett has maintained public silence in recent days, with Likud party chief Netanyahu fuelling speculation his own tenure was about to end in a tweet and video on Friday. "Real Alert," he wrote, warning that a dangerous "left-wing" administration was in the cards.
Yamina announced late on Saturday that Bennett would meet and update its legislators on Sunday, after reports he had agreed to a deal in which he would serve first as prime minister before handing over to centrist Lapid.
A former defence minister, Bennett has reversed course before over ousting Netanyahu, 71, a right-wing leader in power consecutively since 2009 and now on trial on corruption charges that he denies.
With an agreement with Lapid widely reported to have been finalised just before fighting erupted on May 10 between Israel and Gaza militants, Bennett said during the hostilities he was abandoning efforts to form a coalition with the centre and left.
But a ceasefire is holding, a recent wave of street violence in Israel between Arabs and Jews has ebbed, and a Lapid-Bennett partnership could be back on course.
Israeli political commentators, however, were taking nothing for granted.
"The anti-Prime Minister Benjamin Netanyahu government of change is still not an accomplished fact," political columnist Yossi Verter wrote in the left-wing Haaretz newspaper on Sunday.
"It’s premature to pop open the champagne, and also too early to wear sackcloth," he said, questioning whether Yamina's lawmakers could withstand pressure from the right against a deal with Lapid.
If Lapid, 57, fails to announce a government by Wednesday, a fifth Israeli election since April 2019 - a prospect Bennett has said he wants to avoid - is likely.