WASHINGTON, Nov 9 (Alister Bull/Reuters) - The White House sought on Wednesday to limit damage to U.S.-Israel relations following revelations that French President Nicolas Sarkozy called Prime Minister Benjamin Netanyahu of Israel a liar in a private conversation with President Barack Obama.

Our record speaks very clearly about the president's commitment to Israel and he has maintained a very close working relationship with Prime Minister Netanyahu, White House deputy national security adviser Ben Rhodes told reporters, referring to Obama.

Obama has had a difficult relationship with Netanyahu, who criticized him for pushing Israel too hard in the drive for a Middle East peace deal, straining Obama's standing with Jewish American voters as he campaigns for re-election next year.

Reporters covering the G20 summit in Cannes last week overheard French President Nicolas Sarkozy call Netanyahu a liar while talking to Obama.

Instead of contradicting Sarkozy's characterization of Netanyahu, Obama appeared to commiserate. You're fed up with him, but I have to deal with him even more often than you, the U.S. president replied, according to the French interpreter.

The White House declined to comment on the Obama-Sarkozy conversation, citing its private nature.

The exchange was monitored by reporters listening on translation devices while they waited nearby to cover public remarks from the two leaders after their bilateral meeting.

Jewish American voters worried by Obama's views on Israel could hurt him in next year's general election.

They are a powerful voting bloc in Florida, a vital swing state that Obama won in 2008. In September, his Democrats lost a heavily Jewish New York congressional district that had been in the party's hands since the 1920s, with anti-Obama sentiment blamed for the swing.

While not commenting on the liar remark, the White House went out of its way to stress that Obama was fighting for Israel's interests throughout the G20 summit, which was otherwise preoccupied with a euro zone debt crisis.

Our actions speak very loudly, which is that this president has taken strategic cooperation with Israel to unprecedented levels, Rhodes said.

In so far as the Middle East came up at all at the G20, it was the president raising with other leaders his opposition to Palestinian membership in U.N. agencies, he added.

France voted in favor of a Palestinian request to join the U.N. cultural heritage agency UNESCO.

Obama chided him for that vote, according to reporting of their private conversation on a French website that Reuters has confirmed. The day the conversation took place, the Palestinians announced that they would not seek membership of any other U.N. agency.

(Editing by Cynthia Osterman)