The leader of the hardline Likud party added: "I believe that 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 appeared moments later and said the two were still at odds regarding talks with the Palestinians, but both agreed to meet again soon.
She insisted she would not compromise on her position that negotiations with the Palestinian Authority, aimed at 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.
"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.
Pledge on US ties
"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"
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.
"I hope to do so in a unity government," he added.
Lieberman said a Netanyahu-led government would enjoy good relations with Washington.
"Our enemies, unfortunately, are as common as the values and the interests that have united us for all these years," Lieberman said.
"I have no doubt that with Netanyahu's government here we will have good and positive relations with the Obama administration in Washington and with members of congress…" he 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 West Bank Israeli settlement expansion that has irked Palestinian leaders.
Livni's Kadima edged out Likud by one seat in the election, garnering 28 in the 120-seat parliament, but Shimon Peres, the president, invited Netanyahu to form the next government because he appears to have the support of a majority of the elected legislators.
|Lieberman, left, said US-Israel ties will be "positive" with Netanyahu as PM [EPA]
Netanyahu, a former prime minister, has six weeks to form the government and he is expected to extend Kadima a generous offer that includes Livni remaining in her position as foreign minister.
He could alternatively form a narrow coalition with relative ease, giving him 65 seats in the 120-seat parliament, but that would give his smaller coalition partners veto power over major decisions, which could bring down the government in cases of dispute.
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.
Speaking to her Kadima party on Sunday, Livni seemed eager to assume the role of opposition leader.
"The choice is between hope and despair, between promoting and implementing the vision of two states for two peoples and between a lack of direction in that field," she said, adding that if Kadima compromised its platform to join the government it would be "betraying the trust of the public".