KINSHASA- Rebels from the M23 group have gained ground in eastern Democratic Republic of Congo since they attacked two army positions near the border with Uganda and Rwanda on Monday night, a civil society member and a research group said.

Heavy fighting took place throughout the day on Monday as Congolese troops clashed with the group around 50 km (30 miles) northeast of North Kivu's provincial capital, Goma.

A civil society coordinator in the nearby town of Rutshuru, the local administrative seat, said M23 rebels had since moved into the city of Kabindi and were one km away from a land bridge leading into Rutshuru.

"If these enemies manage to dislodge our forces, Rutshuru centre will fall into the hands of the assailants," Jean Damascene Baziyaka told reporters on Tuesday.

The Kivu Security Tracker, a research group that maps unrest in eastern Congo, also reported M23 advances towards Rutshuru with several towns taken along the way.

M23 rebels seized large swathes of territory during an insurrection in 2012 and 2013, before they were chased out by Congolese and United Nations forces into Uganda and Rwanda.

U.N. investigators have previously accused Rwanda and Uganda of supporting the group, which both countries have denied.

Rwanda on Tuesday reiterated that it was not involved in any "belligerent activities" in its vast neighbour after Congo said the M23 were backed by Rwandan soldiers, two of whom it said it had captured.

Uganda Red Cross spokesperson Irene Nakasita meanwhile said 6,000 refugees had crossed the border into Uganda from Congo on Monday.

"They are being taken to a reception centre in Kisoro from where they will be processed and taken to a refugee resettlement camp," Nakasita said.

M23 rebels launched a similar attack in the same areas in November that was rebuffed by the Congolese army.

There have been regional efforts in recent years to get the group to demobilize but its leaders have complained about the slow implementation of a peace accord and accused the Congolese army of waging war against it.

(Reporting by Stanis Bujakera in Kinshasa; Additional reporting by Elias Biryabarema in Kampala and Clement Uwiringiyimana in Kigali; Writing by Hereward Holland and Sofia Christensen; Editing by Nick Macfie)