Rebels in eastern DR Congo have advanced to within a few kilometres of the provincial capital, Goma.
The M23 rebels say they have no plans to take the city but residents fear it could fall at any moment.
The UN Security Council this weekend condemned rebel attacks and demanded an end to outside support for the group, noting they were well equipped.
UK Foreign Minister William Hague has condemned the M23 advance and urged British citizens to leave Goma.
“Any British nationals in Goma should leave, and any in DRC should check the FCO’s updated travel advice,” he said.
Government forces and United Nations troops still control Goma’s airport, but the UN says the humanitarian situation is worsening, with some 60,000 internally displaced people fleeing the fighting.
The UN said more than 10,000 fleeing civilians were seen passing near the airport on Sunday.
The rebels captured the town of Kibumba 30km (19 miles) north of Goma, on Saturday, and have since edged closer towards the North Kivu provincial capital, which lies close to the borders with Rwanda and Uganda.
UN deployed
The UN said its peacekeeping forces, Monuscu, fought advancing rebels throughout Sunday, using rockets, cannon rounds and helicopter gunships.
Although it has deployed 17 quick reaction units on patrols throughout the city, the UN describes the situation in Goma as “extremely tense”.
“There is a real threat that the city could fall into the M23’s hands,” said spokesman Kieran Dwyer.
Monuscu has 270 international staff in Goma, while there are an additional 355 international UN staff in the city.
“We strongly condemn the continued violations of international humanitarian and human rights law by the M23,” said Mr Dwyer.
“The leaders of the M23 must and will be held accountable for their actions.”
The French ambassador to the UN Gerard Araud noted this weekend there would be no solution to the crisis without an agreement between DR Congo and its neighbours, including Rwanda.
Kigali denies allegations that the rebels are backed by Rwandan troops and heavy weaponry.
But UN experts say they have evidence of Rwandan support for the rebels, and this week asked the Security Council to sanction senior Rwandan officials as a result.
The fighting is the most serious since July in the mostly lawless but resource-rich eastern DR Congo.
Nearly 500,000 people have fled their homes since April when the rebels mutinied from the army.
On Tuesday, Uganda closed the Bunagana border crossing near Goma after a request from the DR Congo government, which said the M23 was illegally raising money from people travelling between the two countries.
Last month, a UN panel of experts said Rwanda and Uganda were supplying M23, also known as the Congolese Revolutionary Army, with weapons in what is seen as an on-going battle for control of the region, which is rich in minerals.
Rwanda and Uganda strongly deny the allegation – Rwanda has called on both sides to stop fighting, saying stray bullets have fallen on its side of the border, injuring civilians.
The UN and US imposed a travel ban and asset freeze earlier this week on the group’s leader, Sultani Makenga.
The UN has a large force in DR Congo to help the government establish its authority in the mostly lawless east.