Israeli Prime Minister Benjamin Netanyahu is trying to dissuade his opponents from forming a government as he seeks to hold on to power.
Centrist party leader Yair Lapid has until Wednesday to build a coalition.
Media reports from Israel say he is close to reaching a deal with ultra-nationalist leader Naftali Bennett.
Mr Netanyahu, Israel’s longest-serving PM, fell short of a decisive majority at a general election in March – the fourth in two years.
Mr Lapid could be close to reaching a deal, says BBC Middle East analyst Sebastian Usher.
Mr Netanyahu, who is on trial for fraud, has been in power for 12 years and has dominated Israeli politics for a generation. He was given the first chance at forming a government but was unable to secure the coalition partners needed.
Mr Lapid’s Yesh Atid party came second to Mr Netanyahu’s right-wing Likud at the last election and was given until 2 June to build a coalition.
Our analyst says the outcome of the negotiations depends on whether Mr Bennett, a former defence minister, will agree to share power in what has been described as a “change” government.
However, he cautions that no one should yet count out Mr Netanyahu, 71, whose political survival instincts remain second to none.
Mr Lapid’s 28-day mandate to form a government was interrupted by the recent 11-day conflict in Gaza.
One of his potential coalition partners, the Arab Islamist Raam party, broke off talks because of the violence. There were also clashes in Israeli cities with mixed Arab and Jewish populations.
On Saturday night, Mr Netanyahu’s Likud party made an offer to Mr Bennett and the leader of another potential coalition party to share the premiership in a three-way split.
However, his offer was rejected.
If Mr Lapid is unable to agree a coalition, Israelis could be going back to the polls again later this year.
Under Israel’s electoral system of proportional representation, it is difficult for a single party to gain enough seats to form a government outright. Smaller parties are usually needed to make up the numbers needed for a coalition.