Mr. Netanyahu, the right-wing Israeli leader of the Likud party, is expected to ask Ms. Livni – who heads the centrist Kadima party – just that in a meeting Sunday night as he focuses on forming a new government, something the Israeli president formally asked him to do on Friday.
While Netanyahu has a clear majority of right-wing parties eager to join him following the Feb. 10 election, he is showing a clear preference for a centrist, national unity government that would include Kadima instead of a right-wing coalition.
Netanyahu, a controversial former prime minister, knows that a government without Livni is likely to be viewed as a pariah on the international stage, analysts say. And a hard-line, right-wing cabinet would potentially shackle him and prevent him from taking even the most moderate steps toward peace with the Arab world.
Livni, the foreign minister, won the largest number of seats in the election – Kadima took 28 of the 120 available, compared with Likud's 27 – but found herself unable to muster a sufficient number of political parties that would constitute a governing majority in parliament.
Israeli newspapers here were full of articles of the "tempting" and "generous" offers Netanyahu was to present to Livni in their Sunday meeting, including a proposal that the two party leaders would jointly write government guidelines.
"A real attempt needs to be made to reach a joint position, from within mutual respect and real discussion," Netanyahu said. "It is possible to achieve unity through dialogue and not through dictates and forcing our hand. I have no doubt that whoever sees the state's best interest will place unity as a central goal."
But Livni, meanwhile, had announced over the weekend that she had no interest in joining a rightist government headed by Netanyahu.
"I don't want to be Netanyahu's international stain remover," the foreign minister said last week, adding that she would prefer to be an opposition leader than join a government that would surely bring paralysis to the peace process.
She faces increasing pressure from two camps within the Kadima coalition: left-wing parties such as Labor and Meretz, and other swing voters who left their parties to vote for her earlier this month.
One camp is urging her to be cooperative for the good of the country and join Netanyahu, so as to block the advent of a far-right government. The other is demanding that she stick to her principles and head the opposition rather than sit in a government with ultranationalist parties such as Yisrael Beytenu and smaller parties that represent Jewish settlers in the West Bank.
"Netanyahu is aware that a right-wing religious coalition will expose him to pressure from the West, and that for him it will be better to have a broad coalition," says Shmuel Sandler, an expert on the Israeli political system at Bar-Ilan University, near Tel Aviv. Moreover, Mr. Sandler notes, many of the key figures in Kadima – Livni included – started out in the Likud and bolted in 2005 with Ariel Sharon.
"A lot of the Kadima people are previous Likud people, so for them it's not so hard to go back home," he says. "Most of the Likud people are not as right as they used to be, and would prefer to be influenced by Livni [rather] than the small right-wing parties."
Some of those smaller parties, naturally, are not happy to see Netanyahu working to get Livni in the government.
Kadima joining the government would probably mean that Livni would again have the foreign-affairs portfolio overseeing all negotiations with the Palestinians and the rest of the Arab world. Kadima would probably control at least one other major ministry, such as defense. To bring Livni into the government, Netanyahu would, in principle, have to accept a two-state solution, an Israel next to a state of Palestine, which many of the smaller rightist parties vehemently oppose.
But with Kadima's 28 seats, Netanyahu may no longer need to count on the support of some of the smaller right-wing parties, particularly those representing religious settlers.
This broad-government ideal is also favored by Avigdor Lieberman, the head of the rightist Yisrael Beytenu party – the third largest – which has a secular agenda and would like to see Netanyahu form a government that doesn't have to kowtow to religious parties.
What became clear in recent days was that, although she led the party that won the largest number of votes, Livni is not going to be prime minister. And yet, in the days ahead, the shape of the next Israeli government rests in her hands.
She could decide not to join Netanyahu and stay aligned with a truncated left-wing opposition, citing her principles and the beliefs of those who voted for her. Or, she can make a deal with Netanyahu that will be viewed by some as a sellout, and by others as a mature move for the good of the country – one that will save Israel from lurching radically right.
In a front-page letter to Livni in the Yedioth Ahronoth newspaper, columnist Sima Kadmon captured what seemed to be a dominant theme in Israeli politics in the past few days, indicating the level of pressure on Livni to be a partner in Netanyahu's government-in-the-making. Ms. Kadmon implored Livni to swallow her pride.
"So don't think about either the Right or the Left, Tzipi Livni. Simply think about 7-1/2 million Israelis who are fed up with leaders who say that unity is imperative, but a minute later recant," Kadmon wrote. "Think about what truly is good for them … for them, not for you. For them, not for Kadima."
Following this line of thought, some analysts suggested that Netanyahu will try to invite the Labor Party into a unity government, offering the defense ministry again to Ehud Barak. But other Labor members, particularly the more dovish wing of the party, are dead set against such a move. Daniel Ben Simon, a former journalist and a new member of parliament with Labor, says it won't happen.
"We are going into the opposition with the full conscience of what that means, and we don't think we have the mandate to go to bed with Kadima or the Likud," he said. "Our voters would not support that, so we're going to be an opposition. Livni has a different mandate and she can do what she wants with it."