By Francois Murphy

PARIS, June 28 (Reuters) - Fresh from his success in brokering a deal to overhaul the European Union's institutions, French President Nicolas Sarkozy may soon turn his diplomatic skills to an even tougher challenge -- peace in the Middle East.

Sarkozy is due to meet Palestinian President Mahmoud Abbas, Israeli Foreign Minister Tzipi Livni and Jordan's King Abdullah in the next week, and could decide as a result to help negotiate between Middle East factions, his spokesman said on Thursday.

"If he can be useful, he will do everything he can to be just that. But that will depend on his consultations," David Martinon told a news conference.

A French initiative would be unusual as France is not in itself a member of the Quartet of Middle East mediators -- composed of the United States, the European Union, the United Nations and Russia.

Martinon did not say what kind of action Sarkozy might take, and said the president had not reached a decision.

"He does not particularly want to take initiatives in advance, in abstracto," he said.

"President Sarkozy seems to have the Israeli authorities' confidence and, to my knowledge, he also has the same level of respect from Arab leaders."

Sarkozy has shown himself to be a skilled negotiator, helping to clinch a deal with reticent Poland at a European summit last week when its resistance looked likely to scupper efforts to overhaul EU institutions.

The Islamist group Hamas seized control of the Gaza Strip earlier this month. Abbas responded by sacking the Hamas-led Palestinian government and forming an emergency cabinet in the West Bank, backed by the Quartet and key Arab allies.

That has divided the Palestinian territories into two separate fiefdoms -- a Hamas-controlled Gaza and a Fatah-dominated West Bank.

Sarkozy has friendly relations with former British prime minister Tony Blair, whom the Quartet has made its new envoy in a bid to revive Palestinian-Israeli peacemaking.

Sarkozy's talks with Middle East leaders follow a meeting last week with Saudi Arabia's King Abdullah, which Martinon said were "extremely rich and fruitful".

He will meet Abbas in Paris on Friday, Martinon said, adding: "It will enable President Sarkozy to reiterate to President Abbas all his support in the struggle against extremism and for the reconciliation of the Palestinian people."

Sarkozy meets Jordan's King Abdullah and Israel's Livni separately on Wednesday.

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