It is a question asked more often in hope than expectation: what willBarack Obama do about an Israeli government the US president reportedly regards as acting against the Jewish state's interests, led by a prime minister who is a "political coward"?

As Obama begins his second term there is little sign of the determination to break the deadlock in the Israeli-Palestinian conflict when he took office in 2008, which he then characterised as an open sore afflicting the whole Middle East. Instead, disillusioned former peace negotiators and Middle East policy officials expect his "dysfunctional" and confrontational relationship with Binyamin Netanyahu to stagger along even if the Israeli prime minister returns to power after Tuesday's election with a government even further to the right of the present one.

"It's highly unlikely there's going to be a fundamental shift in policy," said Aaron David Miller, who served six US secretaries of state as an adviser on Arab-Israeli negotiations. "The only thing that would somehow create a new approach on the American side given the domestic travails that the president has, and the preoccupation with Iran and so many other issues, would be [if] a rightwing government [in Israel] begins to take actions that are truly provocative beyond anything that Netanyahu's done. I think [Obama] will go to great lengths to avoid that. Chances are you're going to end up with what you have now, or something that will make any Israeli-Palestinian initiative much less likely."

Obama no long appears to have a plan for the Middle East, said Ghaith al-Omari, Washington-based director of the American Task Force on Palestine and former director of the Palestinian president's international relations department.

"I don't think the administration has figured out what their Israeli policy is right now. You see certain things that will not change. I think there's no appetite here in pressing the Israeli negotiation process," he said.

The already tense relationship between Obama and Netanyahu – which got off to a bad start in 2009 when the president demanded the Israeli prime minister halt construction of Jewish settlements in the occupied territories – is unlikely to improve with the appointment of Chuck Hagel, who has been unusually critical of Israeli policies, as US defence secretary.

Nor will it be helped by the rise in Israel of Naftali Bennett, leader of the rightwing party The Jewish Home, who advocates annexing much of the West Bank and opposes creation of a Palestinian state. He may well join Netanyahu's next coalition and land a place in the Israeli cabinet.

Netanyahu appears determined to go on expanding Jewish settlements even when it causes severe embarrassment to Obama, such as the announcement of construction on a sensitive site outside Jerusalem just after the US was the only major power to back Israel at the UN over recognition of a Palestinian state in November.

The re-election of Obama and Netanyahu's likely return to power has prompted heated political debate in Israel following a column by Jeffrey Goldberg in which the writer, who has close ties with the White House and Israel, said "the president seems to view the prime minister as a political coward" because he is "an essentially unchallenged leader who nevertheless is unwilling to lead or spend political capital to advance the cause of compromise".

The column drew accusations from Netanyahu's Likud party that the White House was interfering in the Israeli election, an echo of his backing of the Republican candidate Mitt Romney in the US presidential contest.

But Netanyahu's rivals swiftly picked up on this to accuse the Israeli leader of jeopardising Israel's most important diplomatic relationship.

A former Israeli prime minister, Ehud Olmert, said Netanyahu was damaging relations with Washington. That charge was echoed by Tzipi Livni, a former foreign minister and leader of the Hatnuah party.

"One can like or not like the president of the United States, but we're talking about Israel's greatest friend and Israel's security, so these remarks should wake up every Israeli," she said.

Miller said he regards the relationship between Obama and Netanyahu as the most dysfunctional he has seen between US and Israeli leaders. But, he added: "It will stagger on because what stands behind it – that is to say the Israeli-US relationship – is too big to fail. Neither Obama nor Netanyahu, no matter how dysfunctional it is, has any desire or rational reason to go to war [with each other]."

But Omari said that doing nothing carries its own risks. "The conventional wisdom is we're going to see a policy of neglect. I'm not sure how sustainable that is," he said.

"What we have seen – and this is problematic not only in terms of US/Israeli relations but US relations with other countries in the Middle East – is there is a sense you can ignore the US and get away with it. This has affected the US's credibility and projection of power. The perceived inability of the US to enforce a settlement freeze, the perceived inability of the US to stop the Palestinians going to the United Nations, and many other such things, basically create a sense that you can say no to the US and it's cost-free.

"The first thing the administration needs to establish – re-establish – is the image of the president as a strong president. If you see any indication of progress in the Middle East it's followed when the US has clearly showed that it's powerful."