TEL AVIV, Jan 23 (Reuters) - Israel believes U.S. President Barack Obama will continue to shun Hamas, in line with a policy set by his predecessor, a top adviser to Prime Minister Ehud Olmert said.

"I don't think that this administration ... will deal with Hamas or will talk with Hamas," the adviser told reporters, a day after Olmert and Obama spoke by telephone.

The adviser, who spoke on condition of anonymity on Thursday, did not say whether Obama had told Olmert explicitly that he would not talk to the Palestinian group which rules the Gaza Strip.

"Talking to Hamas is first and foremost a Palestinian problem. If the international community start talking to Hamas, they will undermine the moderates," the adviser said, referring to Western-backed Palestinian President Mahmoud Abbas.

Hamas, which won an election in 2006, is shunned by the West for refusing to recognise Israel and to renounce violence. The Islamist group became more isolated a year later when it seized control of Gaza after routing Abbas's Fatah forces there.

"This is the fight between the moderates and the extremists in the region and I don't think that anyone has an interest ... to do it and will do it," Olmert's adviser said.

Israel launched a punishing military assault against Hamas militants in Gaza on Dec. 27 with the aim of stamping out years of cross-border rocket attacks. The 22-day offensive ended on Sunday with both sides announcing ceasefires separately.

Obama on Thursday said Hamas should end rocket fire into Israel in order for the ceasefire to last.

Hamas is backed by Syria and Israel's arch foe Iran. Israel considers Hamas a terror organisation. (Reporting by Adam Entous; Writing by Joseph Nasr; Editing by Matthew Jones)

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