By Renee Maltezou, Tom Perry and Lidia Kelly ATHENS/BEIRUT/MOSCOW (Reuters) - The United States has asked Greece to deny Russia the use of its airspace for supply flights to Syria, a Greek official said on Monday, after Washington told Moscow it was deeply concerned by reports of a Russian military build up in Syria. Russian newswire RIA Novosti earlier said Greece had refused the U.S. request, quoting a diplomatic source as saying that Russia was seeking permission to run the flights up to Sept. 24. Russia, which has a naval base in the Syrian port of Tartous, has sent regular flights to Latakia, which it has also used to bring home Russian nationals who want to leave.