The PM urged fellow right-wingers not to back a deal after ultra-nationalist Naftali Bennett said he would form a coalition with a centrist party.
Mr Netanyahu's opponents now have until Wednesday to form a government.
The dramatic announcement by Naftali Bennett, leader of the small hardline Yamina party, set the stage for a series of steps that could push Netanyahu and his dominant Likud party into the opposition in the coming week.
While Bennett and his new partners, headed by opposition leader Yair Lapid, still face some obstacles the sides appeared to be serious about reaching a deal and ending the deadlock that has seen the country hold four elections in the past two years.
Mr Netanyahu, who faces serious corruption charges and could go to jail, fell short of a decisive majority at a general election in March.
If Bennett and Lapid and their other partners can wrap up a deal, it would end, at least for the time being, the record-setting tenure of Netanyahu, the most dominant figure in Israeli politics over the past three decades. Netanyahu has served as prime minister for the past 12 years and also held an earlier term in the late 1990s.
In a televised statement, Netanyahu accused Bennett of betraying the Israeli right-wing and urged nationalist politicians not to join what he called a “leftist government.”
“A government like this is a danger to the security of Israel, and is also a danger to the future of the state,” he said.