Many of the refugees are reported to come from the Buthidaung area in Myanmar's northern Rakhine state, which is 20-25 kilometres east of Maungdaw, Reuters reported.
"We have had big numbers coming across by the day over the six weeks of this emergency. So we are back up to approaching some of those peak arrivals. Clearly we have to be prepared for more arrivals," he said.
"Some said they had fled torching and killings back home; one boy was seen with a big gash across his neck," Edwards added.
Authorities fear an outbreak of cholera could make matters worse as Bangladesh Cox's Bazar has seen several thousand refugees arriving in large numbers with poor facilities in most camps with the local administration struggling to put things in order.
Meanwhile, Myanmar leader Aung San Suu Kyi conducted inter-faith prayers at a stadium in the biggest city of Yangon with Buddhists, Muslims, Hindus and Christians in attendance.
International agencies and governments in western nations have strongly condemned Myanmar's handling of the Rohinga crisis.
The United States had accused Myanmar of "ethnic cleansing" of minority Rohingya Muslims, saying the bloodshed in Rakhine state had shamed Aung Sang Suu Kyi's government.
"This is for peace and stability," party spokesman Aung Shin told Reuters after the inter-faith prayers.