Zimbabwe President Emmerson Mnangagwa said on Tuesday that misconduct by security forces reacting to protests over fuel price hikes would be investigated, but he added the increases were the right thing to do.

"Violence or misconduct by our security forces is unacceptable and a betrayal of the new Zimbabwe," Mnangagwa wrote on Twitter, in his first public comments since returning home from a trip to Russia and other former Soviet countries.

"Misconduct will be investigated. If required, heads will roll," Mnangagwa said, calling for a "national dialogue" over the protests.

Police say three people died during violent demonstrations in Harare and Zimbabwe's second city Bulawayo last week, but human rights groups say evidence suggests at least a dozen were killed. Hundreds have been detained.

Critics accuse Mnangagwa's government of shutting off the internet to prevent a security clampdown from being broadcast to the world.

Jacob Mafume, a spokesperson for the opposition MDC party, said there was an urgent need for dialogue.

"The arrest of labour leaders and opposition members is a familiar script in Africa. We have called for dialogue until our voices have gone hoarse but have been ignored," Mafume said.