MOSCOW, May 23 (Reuters) - Russia hosted representatives of rival Palestinian factions Fateh and Hamas on Monday and praised a power-sharing deal that U.S. President Barack Obama has called an enormous obstacle to Middle East peace.

We very much value your agreement, Russian Foreign Minister Sergei Lavrov told members of Fateh, Hamas and other parties to the deal signed early this month in Cairo.

All peoples need unity, not least the Palestinian people, who are justly seeking a solution to their task of creating a state, Lavrov said.

Israel and the United States have criticized the deal between Palestinian President Mahmoud Abbas's Fateh movement and its Islamist rival Hamas, which they shun.

Obama said on Sunday that the agreement poses an enormous obstacle to peace.

No country can be expected to negotiate with a terrorist organization sworn to its destruction.

A partner of the United States, the EU and the United Nations in the Middle East quartet, Moscow has made a point of calling for the inclusion of Hamas in diplomacy, hosting its leaders and saying isolating it is counterproductive.

Hamas deputy leader Moussa Abu Marzouk said after the talks that Lavrov promised Moscow's support if the Palestinians seek recognition as a U.N. member-state in September. Lavrov did not comment directly on the issue in public.

Obama said the United States would veto a Palestinian statehood bid at the United Nations.

Lavrov also welcomed the Palestinian plans for elections in October. The Palestinian Authority recently postponed the local balloting, which had been scheduled for July, gaining more time to organize voting in the Hamas-controlled Gaza Strip.

Fateh and Hamas have yet to reach a deal setting up a new government, and the choice of a prime minister could help increase Western support for the reconciliation deal.

Marzouk said the factions would hold further talks on candidates early next week and would announce the name of a new prime minister in early June, state-run Russian news agency RIA reported.

(Writing by Steve Gutterman)