A grenade attack in a Nairobi church has killed one person and injured at least 15 others, police in the Kenyan capital say.

"We have been told that the person who threw the grenade was part of the congregation and he fled immediately after throwing it," Joseph Gichangi, the area's deputy police chief, said on Sunday.

Moses Ombati, the deputy police chief for Nairobi, confirmed the number of killed and injured.

The injured were rushed to hospital by other worshippers, police said. Five of those admitted were in critical condition, Ombati said.

Charles Owino, deputy spokesman for the Kenyan police, said the grenade was thrown into the God's House of Miracle Church in the capital's Ngara neighbourhood.

The blast was in a working class area of the city, police and Kenya Red Cross officials said.

"Chairs are thrown about and there is blood on the floor," said Daniel Mutinda, a Kenya Red Cross official at the church, which was constructed mainly using iron sheets.

Glass from windows shattered in the blast, overturned chairs and patches of blood were scattered about the church floor.

A bloodied footstep marked the entrance of the church, which police estimated could accommodate more than 200 people.  Worshippers carried the wounded to cars that rushed them to hospitals, and later huddled speaking in hushed tones.

Members of the congregation said they tried to detain the alleged attacker after the explosion.

"We saw a man running out after the blast and when we chased him he pointed a pistol at us, so we ran back," Kennedy Wasilwa recounted.

Joseph Kimani, who ownes a shop nearby, told the AFP news agency that he rushed to the church after hearing "a loud blast," and initially thought the church had caught fire.

"Then I found people screaming and some were trying to assist others who had injuries," he said.

The incident is the latest in a series of grenade attacks since Kenya sent troops to fight al-Shabab fighters into Somalia in October last year.

The deadliest such blast, on March 10, struck a bus terminal, killing nine people and injuring more than 40 others.