Some 30,000 people have already fled the fighting. The Red Cross reports that 100 refugees a day, mostly women and children, are fleeing across the border into Uganda.
The rebels are led by renegade Gen. Laurent Nkunda, who left the Congolese army and launched his own rebellion after the Democratic Republic of Congo's civil war ended.
He has accused Congo's army of failing to protect his Tutsi people and of collaborating with Hutu militias and ex-Rwandan soldiers who escaped punishment in the 1994 genocide of Tutsis and moderate Hutus in Rwanda. Tutsis are a small minority in Congo, making up 3 percent of the population in east Congo.
Nkunda signed a cease-fire with the government in January, but broke it when he started a fresh offensive in August. He has now proclaimed plans to liberate the entire Congo.
The pro-government Congolese Le Potentiel newspaper based Kinshasa is alleging that two battalions from the Rwandan army are fighting with Nkunda. Rwanda has been accused of backing Nkunda.
In the midst of the crisis, the Spanish general who was recently appointed to run the Congo U.N. peacekeeping mission abruptly resigned.
Unnamed sources told the New York Times that Lt. Gen. Vicente Diaz de Villegas y Herreria said there was not an adequate mandate or resources to complete the peacekeeping mission.
The largest U.N. peacekeeping force -- more than 17,000 troops -- has been present in eastern Congo for several years, but has been unable to stop the violence there. Brig. Gen. Ishmeel Ben Quartey of Ghana will lead the mission for the moment, the U.N. announced.
Amid the fighting, U.N. peacekeepers were planning to evacuate around 50 foreign aid workers from Rutshuru, a town about 60 miles from the eastern provincial capital of Goma, on Tuesday but the attempt failed.
"The aid workers were not able to leave Rutshuru with the planned convoy. The situation is very tense. They were blocked by both the population and soldiers. There are also attacks on humanitarian installations and looting," Col. Delphin Kahimbi, the army's commander of operations in North Kivu, told Reuters.
On Monday, peacekeepers fired into the air at one U.N. compound that was hit by a hail of rocks thrown by civilians protesting the lack of security. City leaders said three people were killed, the Associated Press reported.
U.N. forces were using helicopters and tanks to try to stop the advancing rebels, but said their guerilla tactics made them difficult to locate.
The BBC reported that over the weekend that the rebels captured a major army camp at Rumangabo and the headquarters of Virunga National Park.