"In the end, national sense of responsibility will prevail and we will find a way to join hands for the good of the state of Israel."
Livni said the two were still at odds regarding talks with the Palestinians, insisting that negotiations with the Palestinian Authority, aimed at formally creating a Palestinian state, must continue.
"I see my duty to make sure I hold fast to those principles I presented to the voters and asked for their trust, in this process of forming coalitions as well.
"We did not make progress... this evening did not yield any progress on substance and so we are still apart"
Tzipi Livni on talks with Benyamin Netanyahu
"Therefore this is not a matter of just wording, it's a matter of substance. We did not make progress on that and therefore at Netanyahu's request we will meet again but this evening did not yield any progress on substance and so we are still apart," she said.
While Livni supports the formation of a Palestinian state alongside Israel, Netanyahu does not.
He has championed an "economic peace" with the Palestinians as an alternative and supported the expansion of Israeli settlements in the West Bank - something that has angered Palestinian leaders.
Livni's Kadima won a narrow one-seat victory over Likud in the election, taking 28 of the 120 seats in the Israeli Knesset, or parliament.
However, Shimon Peres, the president, invited Netanyahu to form the next government because he appeared to have the support of a majority of the elected legislators.
'Government of paralysis'
Netanyahu, a former prime minister, has six weeks to form the government and he is expected to offer Kadima a generous coalition deal that includes Livni remaining in her position as foreign minister.
Alternatively, he could form a narrow coalition with relative ease, giving him 65 seats in the 120-seat parliament.
This would, though, grant his smaller coalition partners veto power over major decisions, which could bring down the government in cases of dispute.
|Lieberman, left, said US-Israel ties will be "positive" with Netanyahu as PM [EPA]
Livni has previously said she will join a coalition government only if Netanyahu agrees to a rotation arrangement whereby each would serve as prime minister for half of the government's four-year term, a condition Netanyahu rejects.
After Peres invited Netanyahu to form the next government, Livni said she would refuse to serve as a "fig leaf for a government of paralysis" that did not work to promote peace.
Before meeting Livni on Sunday, Netanyahu also moved to ease concerns that his government would freeze peace efforts and upset relations with the US.
"I intend, and expect, to co-operate with the Obama administration and to try to advance the common goals of peace, security and prosperity for us and our neighbours," he said before meeting visiting US senator Joseph Lieberman.
Lieberman said a Netanyahu-led government would enjoy good relations with Washington.
However, Alexandr Vondra, the Czech prime minister and holder of the European Union presidency, said he expected "a rough start" to Gaza-Israeli truce negotiations in Netanyahu became prime minister.
"I think we can have a bit of a rough start, but we need to move ahead with the peace process because the two-state solution road is narrowing," Vondra said before a meeting of EU foreign ministers on Monday.