An estimated 30,000 Rohingya, a Muslim minority living mostly in Myanmar, have been forced to leave their homes since a bloody crackdown by the army in the western state of Rakhine.
Bangladesh has stepped up patrols on the border to try to stop them from entering, but last week it said thousands had flooded into the country, many with nothing but the clothes they were wearing.
"Based on reports by various humanitarian agencies, we estimate that there could be 10,000 new arrivals in recent weeks," said Vivian Tan, a spokeswoman for the UN refugee agency in Bangkok.
"The situation is fast changing and the actual number could be much higher."
Those interviewed by AFP inside Bangladesh had horrifying stories of gang rape, torture and murder at the hands of Myanmar's security forces.
Analysis of satellite images by Human Rights Watch found hundreds of buildings in Rohingya villages have been razed.
Myanmar has denied allegations of abuse, but has also banned foreign journalists and independent investigators from accessing the area to investigate.
Myanmar's de facto leader Aung San Suu Kyi, a Nobel peace laureate, has faced a growing international backlash for what a UN official has said amounts to a campaign of ethnic cleansing.
On Wednesday she vowed to work for "peace and national reconciliation", saying her country faced many challenges, but did not mention the violence in Rakhine state.
Rohingya community leaders in Bangladesh said another 3,000 displaced Rohingya were stranded on an island in the Naf river that divides the two countries, attempting to enter Bangladesh.
"They have been stuck in the island for almost a week without sufficient food and clothes," Abu Ghalib told AFP.
But a spokesman for the Bangladesh border guards said the claims could not be verified as the island was not Bangladeshi territory.
Bangladesh has reinforced its border posts and deployed coast guard ships in an effort to prevent a fresh influx of refugees.
In the past two weeks, Bangladeshi border guards have prevented hundreds of boats packed with Rohingya women and children from entering the country.
Nevertheless Rohingya leaders in Bangladesh said the number of arrivals had risen this week.
But so far little or no aid has been provided for the new arrivals with Bangladeshi authorities fearing food, medicine and shelter will encourage more to cross the border.
Shinji Kubo, who heads the UN refugee agency in Bangladesh, said the new arrivals needed "urgent" help.
"Obviously these people have come from Myanmar after terrible experiences and without any belongings. The winter is approaching. So everyone is really worried about their wellbeing," he said.
More than 230,000 Rohingya are already living in Bangladesh, most of them illegally, although around 32,000 are formally registered as refugees.
Tan said the UN was urging the Bangladesh government to allow the Rohingya safe haven.
"We are ready to support the government to provide effective humanitarian assistance for these individuals in need of international protection," she said.
Violence in Rakhine -- home to the stateless ethnic group loathed by many of Myanmar's Buddhist majority -- has surged in the last month after security forces poured into the area.
It followed a series of attacks on police posts blamed on local militants.