GENEVA, Sept 11 (Reuters) - The U.N. refugee agency on Tuesday expressed concern about new Syrian visa rules that it said were preventing Iraqis from escaping violence or persecution in their country. Ron Redmond, spokesman for the U.N. High Commissioner for Refugees (UNHCR), said the Syria-Iraq border crossing had become "virtually empty" since Damascus on Monday started enforcing a new policy barring entry to most Iraqi passport holders. "The regulations effectively mean there is no longer a safe place outside for Iraqis fleeing persecution and violence," Redmond told a Geneva news briefing, estimating that 2,000 Iraqis now flee their homes every day. "We are increasingly concerned about their fate as their options for safety are reduced," he said. Iraqis could previously turn up at any Syrian airport or border point and be automatically issued a three-month visa. But Syria, which hosts more than 1.5 million Iraqi refugees, sought to limit entry because it could no longer cope with the influx. Iraqi refugees began moving in huge numbers to Jordan and Syria after the 2003 U.S.-led invasion and toppling of dictator Saddam Hussein unleashed sectarian strife across the country. Jordan tightened its visa policy toward Iraqis two years ago, leaving Syria the only accessible route until this week for thousands of people seeking to flee Iraq. Redmond said the exact details of Syria's new visa regime were not yet clear, though the U.N. agency has received assurances from Damascus that Iraqi refugees currently in Syria would not be forcibly returned.