Congo's presidency, which says neighbouring Rwanda is backing the insurgents, said on X there had been "several" deaths on Thursday and blamed "a foreign army illegally present on Congolese soil".
Neither side offered evidence for their claims.
The incident risked further escalating the war in the mineral-rich east of Democratic Republic of Congo, which has already drawn in several neighbouring countries.
Corneille Nangaa, leader of a rebel alliance that includes M23, said at a press conference that grenades used were the same type as employed by Burundi's army in Congo.
Reuters could not independently verify this.
Burundi has had soldiers in eastern Congo for years, initially to hunt down Burundian rebels there, but more recently, to aid in the fight against M23.
Burundian army spokesperson Brigadier General Gaspard Baratuza condemned the attack. He said on X there were no Burundian soldiers in Bukavu, but did not specifically address Nangaa's claim about the grenades.
The rebels have been advancing through the east of the country since the start of the year, seizing Bukavu and the region's largest city Goma.
After the blast in Bukavu, people ran through the streets, some carrying the wounded.
A medical source said 68 people were being treated for injuries at the city's general hospital.
Witness Musanga Tambwe described hearing two loud explosions.
"A plastic bag was thrown in front of us... We didn't know what was inside. We were just looking at it. Suddenly, there was an explosion. Poof! We fell to the ground and ran away," said Tambwe, who was later treated for wounds at the hospital.
Nangaa said that he was not wounded and other senior members of the rebel grouping were safe.
Congo, the UN and Western powers say neighbouring Rwanda is backing M23 - accusations Rwanda denies.
Rwanda has said it is defending itself against the threat from a Hutu militia, which it says is fighting with the Congolese military.
M23 has been trying to demonstrate that it can restore order in the territory it has captured from Congo's army, and has re-opened ports and schools.
This M23 advance is the gravest escalation in more than a decade of the long-running conflict in eastern Congo, rooted in the spillover of Rwanda's 1994 genocide into Congo and the struggle for control of Congo's vast mineral resources.
