The violence comes despite parallel peace attempts backed by the United States and Qatar and after the late June signing of a peace accord between Kinshasa and neighbouring Rwanda.
Eastern Democratic Republic of Congo, bordering Rwanda and rich in resources, has endured three decades of conflict.
Violence spiked between January and February with the Rwandan-backed M23's capture of the major cities of Goma and Bukavu.
The M23 and the Congolese army accused each other in weekend statements of "trampling" on peace efforts or "violating" the accord's principles.
The M23 now insists it must resort to "self-defence", while the army says it reserves the "right to retaliate".
Recent months have seen a fragile lull in the provinces of North and South Kivu. But NGOs and the United Nations have denounced violations against civilians including murder, kidnap and rape in M23-controlled areas.
The army in a statement Saturday pointed to what it said were several attacks by the M23 on its positions Friday and Saturday in the two provinces.
The M23 responded by saying the army had carried out drone strikes on several localities.
Sunday saw the group take control of Nzibira, a gold mining town some 80 kilometres (50 miles) southwest of Bukavu, local and security sources said.
Nzibira "is in M23 hands after intense fighting Sunday against wazalendo fighters", a civil society official told AFP on condition of anonymity, using a term for local militia backing the armed forces.
An army source confirmed that account.
The fighters who entered the town Sunday "do not disturb anyone and don't speak to us. We can see them through the window", said a nurse by telephone from a Nzibira hospital.
Observers now expect an M23 offensive to follow targeting Uvira, a South Kivu city of some 500,000 people opposite the Burundian economic capital Bujumbura on the other side of Lake Tanganyika.
The city is currently under Congolese army and pro-Kinshasa militia control.
Security forces say Burundi's military has shored up its presence at the border.
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