A spokesman for the group said the assault was punctuated by heavy gunfire.
Colonel Vianney Kazarama, spokesman for the M23 rebel movement, said the rebels are now fighting for control of the rest of Goma.
"We already took the airport," he said. "We are now inside the city of Goma."
Goma was last threatened by rebels in 2008 when fighters stopped just short of the provincial capital.
The United Nations has said that if Goma was to fall, a humanitarian catastrophe would result.
The rebels are opposed by Congolese government forces backed by UN peacekeeping troops and their attack helicopters.