Footages from the scene showed a crowd gathering at the Huruma residential estate as rescuers dug for survivors.
At least seven people are known to have died.
The Red Cross criticized “chaotic scenes” as rescuers arrived.
At least 121 people had been rescued, police told Kenya’s KTN network .
It is unclear how many people are trapped beneath seven floors of concrete.
The rainfall in addition has caused landslides, washed away houses and flooded roads.
Fourteen people died in the Nairobi rains, including those in the collapse, police told KTN. Another four died when a wall toppled over, officials said.
The Huruma neighborhood is a poor district on the outskirts of Nairobi made up of narrow streets, meaning firefighters struggled to get to the scene and were delayed by large crowds.
After some time, the army took charge of the rescue – with the help of the Kenyan Red Cross.
Residents said that the building shook violently in the rain before collapsing.
Poor building standards are a fact of life in Kenya, correspondents say. A survey carried out last year found that more than half the buildings in the capital were unfit for habitation.
The high demand for housing in Nairobi has led to some property developers bypassing building regulations to reduce costs and increase profits.
There has however been some good news for the rescuers who extracted a number of children from the wreckage.
In 2015, President Uhuru Kenyatta ordered an audit of all the buildings in Kenya after a spate of collapses.
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