The incident occurred in a very poor neighbourhood near Nairobi's international airport southeast of the capital, Red Cross spokeswoman Noellah Musundi told AFP.
"The information is still scanty (about) the exact number of missing people, but at this stage, a mother and her three children are missing," she said.
Police and local residents said the authorities had been alerted on Monday evening after cracks appeared in the building, prompting an urgent evacuation before it collapsed two hours later, at around 10:00 pm (1900 GMT).
"The collapse was not completely unexpected, which allowed us to evacuate most of the people," said Musundi of the incident which occurred in the Kware area of Mukuru Kwa Reuben, one of the biggest slums in Nairobi.
According to Pius Masai, a spokesman for the National Disaster Management Unit, 128 tenants had been rescued and accounted for by midday.
"Most families cooperated and (were) evacuated safely. However, it is believed that some people may have been trapped. Rescue efforts are ongoing," he said.
He said the rescue operation was difficult since space was limited and the adjacent seven-storey building also appeared unstable.
"It is a very delicate operation but we hope to finish it safely," he said.
Tenants in the adjacent building were asked to evacuate.
No planning permission
Images posted on Twitter by the Kenyan Red Cross showed work to clear the rubble with the help of an excavator, with parts of the building's red roof clearly visible.
A fire engine was also on site as well as a number of soldiers.
Masai also appealed for anyone with cutters or drilling equipment to join the search and rescue operation.
Quoting City Hall, The Star newspaper said the structure was built in 2007 without planning permission or approval.
"Kware area was unplanned. No developments are allowed there. But you find that most of these developers were brought by politicians," Nairobi Lands executive Christopher Khaemba told the paper.
Police are reportedly looking for the owner of the building.
Several buildings have collapsed in recent years in Nairobi and other Kenyan cities, where a property boom has seen buildings shoot up at speed, often with little regard for regulations.
Such incidents have raised questions about the quality of building materials and construction standards in a country where rampant corruption has seen unscrupulous developers using bribes to avoid regulations.
In April 2016, 49 people died when a six-storey building collapsed in a poor neighbourhood northeast of the capital following days of heavy rain which caused floods and landslides.
The building, constructed two years earlier, had been slated for demolition after being declared structurally unsound.
Although around 150 families were living there in tightly-packed conditions, an order to evacuate the building and demolish it was ignored.