One image circulated among Israeli critics of Benjamin Netanyahu, the right-wing Israeli prime minister, summed up how they hoped the US election would influence their own country's ballot in January.

At the top was a photo from Tuesday that has since gone viral on the web of a smiling Barack Obama, the US president who had won re-election, hugging his wife Michelle, with the caption "4 more years". At the bottom was a picture of a grim-looking Mr Netanyahu with the caption: "3 more months."

The image reflected widespread speculation by Israeli commentators that Mr Netanyahu's campaign for a third premiership may be hurt by the victory of Mr Obama, perceived as being critical of the Israeli premier's hard-line policies towards the Palestinians and Iran.

Mr Netanyahu had also appeared to have supported Mr Obama's Republican challenger, Mitt Romney, with whom he shares a network of powerful friends and a key financial backer, the US billionaire Sheldon Adelson.

Israeli analysts said yesterday that Mr Netanyahu was likely to work hard in coming weeks to repair his relationship with Mr Obama in a bid to avoid losing popularity among an Israeli electorate that values Israel's ties with the US, the country's staunchest ally.

Yaron Ezrahi, an Israeli political expert, said: "Israelis respond very seriously to a decline in American support. The Israeli public recognises the US's vitality to Israel and the fact that Netanyahu has corrupted his relationship with the White House."

Israeli media this week also speculated that Mr Obama's win may encourage prominent figures on Israel's centre and left - such as Ehud Olmert or Tzipi Livni, a former premier and an ex-foreign minister, respectively - to challenge Mr Netanyahu for the premiership.

Indeed, rivals like Mr Olmert have been quick to highlight Mr Netanyahu's betting on the wrong horse in the US presidential race and condemn his apparent intervention in Washington's election.

Mr Olmert, speaking to Jewish community leaders in New York this week, attacked Mr Netanyahu's support for Mr Romney, saying that "what took place this time was a breach of all the rules, when our prime minister intervened in the US election in the name of an American billionaire". He was referring to Mr Adelson.

The Israeli leader was believed to be backing Mr Romney partly due to expectations that the Republican politician would apply very little pressure on Israel to curb its Jewish settlement expansion in the occupied West Bank, and adopt a harder line towards Iran's nuclear programme and support Israel's stances in peace talks with the Palestinians.

Some satirical images circulated by Israelis on social-networking sites suggested that Mr Obama may seek revenge for Mr Netanyahu supporting his opponent by himself interfering in Israel's election.

One image attached a Hebrew-language dialogue in speech bubbles to photographs of the two leaders, in which Mr Obama asks Mr Netanyahu, "Say, do you remember how much you intervened and tried to influence the presidential elections here in the States?" When Mr Netanyahu replies: "Yeah, why?" Mr Obama says, "Oh, no reason."

Nevertheless, Mr Netanyahu yesterday rejected reports that he has a tense relationship with Mr Obama. Speaking to reporters at a hospital ceremony in the southern city of Ashdod, he said: "I hear voices in the past few days - on the Israeli side - trying to spark a dispute between Israel and the US. This isn't going to help them. The partnership between Israel and the US is stronger than ever and I'll continue strengthening it."

Despite such statements, Mr Netanyahu's supporters still appear to be campaigning against Mr Obama's policies in a possible bid to minimise any political damage the premier may suffer from Mr Romney's loss.

Yisrael Hayom, a free Israeli newspaper funded by Mr Adelson and known as the "Bibiton" - a play on Mr Netanyahu's nickname, Bibi - because of its open backing for the prime minister, yesterday warned against the economic consequences of Mr Obama's election.

Yesterday, one article headlined: "America Chose Socialism" warned that Mr Obama's second term "could lead to a bigger economic crisis than that of 2008".

Palestinian leaders yesterday also expressed optimism that Mr Obama's victory could hurt Mr Netanyahu's re-election bid.

Nabil Shaath, a senior official of the secular Fatah movement that holds sway in the Israeli-occupied West Bank, said: "Obama isn't [Muslim leader] Saladin, but he's preferable over Richard Lionheart Romney, who doesn't know anything about the Palestinian issue."

Mr Shaath compared Mr Romney to the 12th Century English king, Richard I, known as the Lionheart, who had led the Third Christian Crusade against Muslims and had scored numerous victories against Saladin.