JERUSALEM, Dec 19 (Reuters) - Israeli Prime Minister Benjamin Netanyahu said on Sunday his government would pursue Middle East peace efforts despite the collapse of US-brokered direct negotiations with the Palestinians.

Netanyahu's comments to American Jewish activists followed similar remarks by Palestinian Prime Minister Salam Fayyad who told an Israeli television station the sides should not give up and ought to move forward to resolve core conflict issues.

Face-to-face peace talks broke down this month after Washington formally gave up its efforts to persuade Israel to temporarily halt Jewish settlement building in occupied land, an issue over which the talks reached an impasse in September.

In a speech to Jewish activists at a forum on combating world criticism of Israel, Netanyahu said: The quest for peace is important, and my government shall continue to move toward this goal. We want peace, because we don't want war.

He vowed to fight what he called attacks on our legitimacy, alluding to critics whose denunciations Israel sees as challenging its right to exist as a Jewish state.

Fayyad, in a rare interview with an Israeli television station, said in remarks broadcast on Saturday that he thought it was possible to get this process to move forward despite the impasse and urged new talks focused on key conflict issues.

We should not be discouraged because we have failed before, we should not give up, Fayyad told Israel's Channel 2.

George Mitchell, the US Middle East envoy, pledged in a brief round of shuttle diplomacy last week to strive for progress despite the setback in meeting President Barack Obama's goal of a two-state solution deal between Israelis and Palestinians by late next year.

Two of Mitchell's deputies have now arrived in the region to explore ways to resume negotiations, Israeli media reports said.

Palestinians have insisted Israel freeze settlement building they see as depriving them of land they seek for a state before negotiations may resume. Israel rejects this demand as a precondition for holding talks.