"Bakytzhan Sagintayev came under fire from terrorists in Almaty, rioters tried to kill him," Babakumarov said in a statement, published on Telegram.
Earlier on Sunday, Kazakh news portal orda.kz reported on the mayor assassination attempt, which killed the driver and left several people injured.
A wave of protests swept across Kazakhstan earlier this week, following a sharp rise in gas prices. Despite the government's attempts to quell the discontent and promises to bring the prices down, protests turned violent and descended into clashes with law enforcement officers in several regions.
The government declared a state of emergency until 19 January. Kazakh President Kassym-Jomart Tokayev asked for the Collective Security Treaty Organization's (CSTO) assistance, which was granted. CSTO peacekeepers were sent to Kazakhstan.
The Kazakh president said Friday that there were still militants who continued resistance and pledged to fight those who do not lay down their arms. At the same time, Tokayev has said that the government had reached a compromise with peaceful protesters on urgent social and economic issues.