(Adds background, paragraphs 11-13) By Arshad Mohammed WASHINGTON, Jan 10 (Reuters) - If U.S. President George W. Bush launches a major new push to promote Israeli-Palestinian peace, he could hardly have picked a less favorable time. The Palestinians are riven by factional conflict between President Mahmoud Abbas of Fatah and the Hamas-led government of Prime Minister Ismail Haniyeh. With their power struggle at times flaring into armed conflict between rival militants, it is unclear who might speak for the Palestinians or have the support to negotiate a deal. On the Israeli side, Prime Minister Ehud Olmert remains politically weak and relatively unpopular in the aftermath of last summer's 34-day war with Hezbollah in Lebanon. Without strong leaders on either side, analysts wonder whether a U.S. push -- if one comes -- will go anywhere. While Bush's attention has been firmly focused on whether to change policy on Iraq, there have been hints the White House is considering taking a new tack on the Israeli-Palestinian issue after what critics regard as six years of neglect. "The Bush administration, for the first time, seems to be contemplating a serious effort to deal with the core issues of the conflict," Dennis Ross, a U.S. peace mediator under former Presidents George Bush and Bill Clinton, wrote this week. "One might ask if trying to address the core issues is appropriate at a moment when Palestinians are locked in an internal stalemate and the Israeli public lacks confidence in its government," he said in a New York Times opinion piece. U.S. Secretary of State Condoleezza Rice leaves on Friday on a weeklong trip to the region that will include visits to Jerusalem, Ramallah, Luxor, Egypt, Amman, Kuwait City and Riyadh as well as stops in London and Berlin. U.S. officials have sought to play down expectations for the trip, Rice's eighth to the region during her two years as secretary of state, and have suggested she is testing the waters to see what may be possible. PRESSURE FROM ALLIES, IRAQ STUDY GROUP The White House is under pressure from its European and Arab allies as well as the bipartisan Iraq Study Group to take a more active role to try to resolve the Arab-Israeli problem despite the divisions among the Palestinians. Abbas favors a two-state peace deal between Israel and a Palestinian state in Gaza, the West Bank and East Jerusalem, while the Hamas-led government remains officially committed to the destruction of Israel. Hamas' exiled leader, Khaled Meshaal, appeared to soften that position on Wednesday, telling Reuters that Hamas acknowledged the existence of Israel as a reality but formal recognition would only be considered when a Palestinian state was created. Bush's speech on his new Iraq policy on Wednesday night may provide clues as to how seriously the administration plans to pursue the peace process. While skeptical the timing is right, some analysts also questioned whether the administration, consumed by the violence in Iraq, had the stomach to tackle Arab-Israeli peace. "I just don't see that the timing is propitious and I don't see that the president is solidly behind it," said Edward Walker, a former U.S. ambassador to Israel and Egypt, suggesting the administration may be more interested in deflecting criticism than in embarking on a major effort. Aaron Miller, a veteran U.S. peace negotiator now at the Woodrow Wilson International Center think tank in Washington, said the odds of success "are very long" but argued the problems only get worse without U.S. attention. He also made the case that the weak leaders on both sides and the growing power of non-state actors like Hezbollah in Lebanon made the problem harder to solve since the last major U.S. effort ended in failure in 2000. "The challenge that we face now, as opposed to the ones we faced six years ago with Clinton, is now galactically more complex," he said. "You could argue that is precisely why it's worth trying to manage. Urgency can create a certain measure of desperation and opportunity."