Shimon Peres, the Israeli president, is edging towards inviting the Likud leader, Benjamin Netanyahu, to form a government, and some Kadima members have sworn they would reject any accord that would put their leader, foreign minister Tzipi Livni, under Mr Netanyahu in a new Israeli cabinet.
In these circumstances, an American intervention could be vital. Sources in Washington who have discussed the situation with members of Mr Obama's team now believe that the new president is prepared to play a role behind the scenes to ensure that an alliance can emerge.
"His team is prepared to take the plunge privately if it is never made public," said a well-placed source in Washington.
The Obama administration insists publicly that it will work with whatever coalition government Israeli politicians come up with. The president's advisers are reluctant to be seen meddling in the messy business of constructing Israeli governments.
But they also believe Mr Obama could wield his political capital to ease the birth of a centrist government if one of the main players asks him to intervene.
Mr Netanyahu - widely known as Bibi - is considering offering to share power with Kadima and may need Mr Obama's help to press Mrs Livni to take her party into a new government with Mr Netanyahu as prime minister.
"Bibi will not want a hard Right government," said the source, who is well acquainted with both US and Israeli politics. "He might try to enlist the Americans to lean on Kadima to join his government."
Mr Netanyahu is still haunted by the failure of his last stint as prime minister in the 1990s, when he was forced to the Right by the small hardline parties who propped up his coalition government.
"Our priority is a wide national unity government with both the Left and Right together," Yuval Steinitz, a confidante of Mr Netanyahu told The Sunday Telegraph. "But it's up to Kadima to decide whether they prefer that wisdom or their pride prevails."
For Mr Obama, there could also be an advantage to helping forge the new coalition. In exchange for US assistance in bringing the main players together, the US President would gain leverage to launch his own plan for Middle East peace, rather than getting bogged down in endless negotiations based on Israeli priorities.
But it is not clear whether Mrs Livni would respond positively to overtures from Mr Netanyahu or pressure from Mr Obama.
Middle East experts in Washington believe that Mrs Livni is under pressure from some in her party to stay out of a Netanyahu government in the hope that it fails and clears the way for her Kadima party to return to power.
A source, who has just returned from Israel, said: "There are those saying to her: 'Let Bibi rot. Let him have a government with all the right wingers. It won't last, it will implode; you'll look great and you'll be set fair.'
"Others are saying that governments can struggle through and we've never tested our party to see whether it has enough glue to hold together. Bibi will chip away at your supporters to try to break them away to form a new faction. So we should join the government now."
The danger of a schism in Kadima, founded by Ariel Sharon in 2005 as an alliance between like-minded Labour and Likud politicians, is being openly discussed. "It seems that Kadima will split if Livni refuses to go into coalition," said Mr Steinitz.
Many within the Obama administration think Mrs Livni would be better advised to help form a practical government of national unity now, since that would give Palestinian leaders a partner for peace with whom to deal. They point to the fact that as a former member of Mr Netanyahu's Likud party, she has less far to travel politically than had the members of some previous Israeli cross-party pacts.
Similarly if Mrs Livni is asked to form a government, they believe it would be best if she was prepared to offer Mr Netanyahu an alliance and for him to serve under her.
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