JERUSALEM, Oct 12 (Reuters) - Middle East envoy Tony Blair said on Friday that, despite their apparent weakness at home, President Mahmoud Abbas and Prime Minister Ehud Olmert can deliver a Palestinian-Israeli peace deal.

The former British prime minister told Israel's Yedioth Ahronoth newspaper in an interview that both leaders would be strengthened if they managed to reach a deal at a U.S.-sponsored peace conference near Washington next month.

"People say to me the status of both is too weak to reach a deal, but to my mind that is not the issue," Blair said in remarks published in Hebrew.

Since last year's Lebanon war against Hezbollah guerrillas, Olmert's popularity ratings have badly lagged behind his rivals, while Abbas was weakened by the loss of the Gaza Strip to Hamas Islamists in June.

Blair said: "Olmert is endowed with courage and intelligence. The question is whether he can deliver a viable solution that has peace and security. If he can, his position is strong because if you ask Israelis if they want peace, they will answer positively."

"With (Abbas) there is the same equation. I have spoken to many Palestinians in recent months and I have no doubt that they want a leader who will lead them to their own state.

"If he is able to show his people that there is a realistic possibility to reach statehood, he will be strong."

Blair is tasked by the Quartet of Middle East mediators -- The United States, Russia, the European Union and the United Nations -- with helping the Palestinians on economic and security issues.

Blair warned Israel that failure to reach a deal with the current Palestinian leadership would leave the conflict unresolved for many years.

"If you cannot reach a deal with the current Palestinian leadership ... then the Palestinian with whom you will be able to reach an agreement has not yet been born," Blair said.

He said that he took on the role of helping resolve the conflict in June because of "my sense of a mission to resolve what I believe is the most important conflict in the world at the beginning of the 21st century."

He said he had no fear of failure: "There are people who think I am mad that I chose to endanger my reputation in a place which delivers only failures. But between ourselves, what's point of being a politician if you don't take risks?"

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