The U.N. agency has been counting those entering Syria in recent months, however, and has found an average of 2,000 a day leaving Iraq by that route.
The outflow of refugees caught the agency by surprise because it was hard to detect.
``If people flee to camps, it's quite visible,'' Redmond said. ``This is a steady stream of people now who are leaving.''
The Jordanian government says another 1,000 a day are entering Jordan, he said.
White House spokesman Tony Fratto said he couldn't comment on whether the figures were accurate.
``Obviously there has been some level of violence in Baghdad and it's natural that you would expect some people to leave,'' Fratto said. ``But we're going to continue to work to bring down violence so that Iraqis can stay there and stay in a free country and a free democracy.''
Redmond said almost all the refugees are believed to be staying in Syria and Jordan, but that a few are returning to Iraq and still others are going to other countries. He noted that Iraqis were the leading nationality seeking asylum in Europe in the first half of this year, with 8,100 applying - a 50 percent increase over the year-earlier period.
Figures were unavailable for arrivals in other neighboring countries, but the agency says an additional 50,000 Iraqis a month are fleeing their homes but remaining within Iraq, which classifies them as ``internally displaced'' rather than as refugees who have crossed an international border.
``We've got a displacement crisis under way here, and the international community needs to do more to chip in to support the humanitarian needs of these people,'' Redmond said.
The influx has been driving up prices in Syria and Jordan for housing, food and other commodities, Redmond said.
UNHCR estimates that 425,000 Iraqis have been displaced this year alone, largely due to sectarian violence sparked by the bombing of an important Shiite shrine in the city of Samarra in February,
Redmond recalled that, before the 2003 invasion, UNHCR had made contingency plans for 600,000 refugees and displaced people, and had a budget of $154 million. But there was no mass exodus in the beginning, and UNHCR has long since scaled back to $29 million.
UNHCR now estimates that 1.8 million Iraqis are living in neighboring countries and 1.6 million are displaced internally, but those numbers include many who fled during the 1990s, long before the invasion, Redmond said. As of July, Iraq's population was estimated at 26.7 million.