On Wednesday, M23 fighters launched an offensive in the settlements Kirotshe and Shasha, on the shores of Lake Kivu, about 30 kilometers west of Goma.
Clodius Buhaze, a local civil society figure, said that fighting had been ongoing since the morning, with M23 rebels gaining ground on Congolese troops as well as fighters from enemy militias.
Several witnesses told AFP that soldiers abandoned their positions in Kirotshe in the morning before returning to the fight in the afternoon.
Clashes with troops erupted on Monday in North Kivu province - where the Tutsi-led M23 have captured swathes of territory since re-emerging from dormancy in late 2021 - and continued through Wednesday.
The latest clashes tighten the vice that is closing around Goma, a city that lies on the border, with Rwanda to its east and Lake Kivu to its south.
The M23 is in control of much of the territory north of Goma, and is encroaching from the west, threatening to cut off all road links.
On March 3, the Angolan presidency - which is mediating the conflict for the African Union - announced a ceasefire for eastern DRC that was supposed to take effect at 1000GMT on Tuesday.
The failed ceasefire follows several other unsuccessful regional peace initiatives.