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DUBAI, Oct 12 (Reuters) - Moscow's military intervention in Syria will have "dangerous consequences", escalating the war there and inspiring militants from around the world to join in, senior Saudi Arabian officials told Russia's leaders on Sunday, a Saudi source said.

Saudi Arabia will continue to strengthen and support the moderate opposition in Syria, the source said, citing positions outlined by Defence Minister Mohammed bin Salman and Foreign Minister Adel al-Jubeir in their meetings in Russia with President Vladimir Putin and Foreign Minister Sergei Lavrov.

"The Russian intervention in Syria will engage them in a sectarian war," the source said on Monday.

"The recent escalation will contribute in attracting extremists and jihadists to the war in Syria," the source said, adding that the Kremlin's actions would also alienate ordinary Sunni Muslims around the world.

The Saudis urged Russia to help fight terrorism in Syria by joining the existing coalition comprising more than 20 nations that is battling Islamic State militants, the source said.

He also reiterated that Syrian President Bashar al-Assad must quit as part of a process agreed at a Syrian peace conference held in Geneva in June 2012.

"Assad should leave and the Saudis will continue strengthening and supporting the moderate opposition in Syria," the source said.

Moscow's intervention has infuriated Assad's regional foes, including Saudi Arabia, who say Russian air strikes have been hitting rebel groups opposed to the Syrian leader and not just the Islamic State fighters Moscow says it is targeting.

(Reporting by William Maclean; Editing by Gareth Jones) ((william.maclean@thomsonreuters.com ; +97143664253; Reuters Messaging: william.maclean.thomsonreuters.com@reuters.net: Twitter: @WMacleanR))