The CNDP says it is battling for the rights of DR Congo's Tutsi population, while the country's government says that the rebels are supported by neighbouring Rwanda.
Kigali denies any involvement with the rebels, saying that the fighting in DR Congo is a problem for Kinshasha to address alone.
Kanyabayonga is a strategically important town for the rebels as they aim to win overall control of North Kivu province.
Lying about 175km of Goma, Nord-Kivu's capital, it is where the province's main roads intersect.
Whoever controls the town will have a stranglehold on supply routes across North Kivu.
Jean-Paul Dietrich, military spokesman for Monuc, the UN mission in DR Congo, said that Congolese troops had deployed in large numbers to defend Kanyabayonga.
"As far as the FARDC [government forces] are concerned, we're talking about several hundred men," he said.
But the CNDP said that their fighters had made their way to the edge of the town without meeting any resistance from government troops.
"The situation is calm," Bisimwa said. "We can see that the adversary is still fleeing."
Over the last two weeks, Nkunda's fighters have surrounded Goma, prompting UN peacekeepers to send reinforcements to protect the city.
About half a million people live in Goma and there are fears that civilians could bear the brunt of violence should the clashes between the CNDP and government forces reach the city.
Hundreds of people are already estimated to have been killed amid the fighting, while thousands of civilians have fled their homes to overcrowded refugee camps administered by the UN mission.
|Refugees inside a camp at Kiwanja are still under direct threat, the UN says [AFP]
The UN said on Thursday that about 20,000 people living in the Kibati camp, a few miles north of Goma, will soon be moved for security reasons.
"Kibati is a disaster. You can't have people there, so close to the front line," David Nthengwe, a spokesman for the UN High Commissioner for Refugees (UNHCR), said.
"The military are always going through the camp, in and out, and knowing the behaviour of the army, anything can happen.
"Some will not move because they will fear they are being moved further away from their homes. So that will be a factor to consider."
On Wednesday, Monuc said that CNDP fighters had forced thousands of displaced people to run away from a refugee camp at Kiwanja, around 80km north of Goma.
Human rights monitors have said Nkunda's fighters and pro-government Mai Mai fighters have killed innocent civilians amid fighting to win control of Kiwanja.