The building, which was under construction in Kiambu, caved in on Monday morning, the town's Governor Kimani Wamatangi said on Twitter.
"We have lost five people in the collapsed... building," he said, adding that search and rescue efforts were under way.
"Several people have already been pulled out of the rubble, and rushed to the hospital. Sadly, some are feared to have succumbed to their injuries," he said.
Emergency workers and volunteers attempted to help people trapped under heavy concrete and brick debris. A married couple as well as a mother and her two children among the five fatalities listed by the governor.
The cause of the collapse was not immediately clear.
In the past, shoddy construction and flouted regulations have led to deadly accidents in Kenya.
The East African nation is undergoing a construction boom, but corruption has allowed contractors to cut corners or bypass regulations.
At least three people died in December 2019 when a residential building collapsed in Nairobi.
That incident came three months after seven children died and scores were injured when their school was flattened in an accident blamed on third-rate construction.
In April 2016, 49 people were killed when a six-floor apartment building crumbled in the northeast of the capital after days of heavy rain caused floods and landslides.
The building, constructed two years earlier, had been scheduled to be demolished after being declared structurally unsafe.