He had earlier Sunday tried to cling to power by offering his own, last-ditch power-sharing agreement to several former allies including Bennett.
He warned Israel would otherwise be ruled by a dangerous “left-wing” alliance.
Lapid has until Wednesday 11:59 pm local time (2059 GMT) to build a coalition of at least 61 deputies, a majority in the 120-seat Knesset.
A Lapid government would also include the centrist Blue and White party of Netanyahu’s rival Benny Gantz and the hawkish New Hope party of his former ally Gideon Saar.
Avigdor Lieberman’s pro-settlement Yisrael Beitenu party as well as historically powerful Labour and the dovish Meretz party would also join.
The shaky arrangement would need the backing of some Arab-Israeli lawmakers of Palestinian descent in order to pass a confirmation vote in parliament.
The intense talks follow weeks of escalating tensions between Israel and the Palestinians, including a deadly 11-day exchange of rocket fire from Gaza and devastating Israeli airstrikes.
The war with Hamas that ended with a 21 May truce, as well as violence in the occupied West Bank and in mixed Jewish-Arab towns in Israel, initially appeared to leave Netanyahu more likely to hold onto power.
But political scientist Gayil Talshir at Hebrew University told AFP on Sunday that Israel was now “closer than ever” to a coalition of change, adding: “Netanyahu is in a desperate position”.
Netanyahu’s Likud party won 30 seats in the March elections but failed to form a governing coalition after his far-right partners refused to sit with Arab factions or receive their support.
Lapid, whose party won 17 seats, was then given four weeks to form a government.
Netanyahu had previously pushed for yet another election—Israel’s fifth in a little more than two years.
On Sunday Netanyahu offered his own proposal of a rotation agreement with Bennett and Saar. But Saar on Twitter said he remained committed to “replacing the Netanyahu regime”.
Netanyahu in a video then called on Saar and Bennett to “come now, immediately” to meet him and join a three-way rotation government, warning they were “in crucial moment for the security, character and future of the state of Israel”.
Lapid’s “change” coalition also still faced several obstacles.
Some right-wing lawmakers object to a partnership with politicians from Israel’s Arab minority, around a fifth of the population.
The recent Gaza conflict sparked inter-communal clashes between Jewish and Arab Israelis in mixed cities.
Arab politicians have also been divided about joining a government headed by Bennett, who supports expanding Israeli settlements in the occupied West Bank, where Palestinians hope to create a state.
Even with support from an Arab party, a new coalition in Israel is unlikely to reverse years of Israeli settlement construction or bring peace any time soon with Hamas in Gaza.
If the anti-Netanyahu camp does not manage to form a government on time, a majority of 61 lawmakers could vote to ask the president to name a new premier.
Another scenario would see the country return, yet again, to the polls.