A grenade set off by a worshipper in a Nairobi church killed one person on Sunday and injured at least 10 others, police in the Kenyan capital said.

"We have one fatality, and there are more than 10 others who have been injured," area deputy police Chief Joseph Gichangi said.

"We have been told that the person who threw the grenade was part of the congregation and he fled immediately after throwing it," he added.

The attack targeted the God House of Miracle Church in Nairobi's Ngara neighbourhood.

The Kenyan capital has been hit by a series of unclaimed attacks since late 2011. The deadliest blast, on March 10, struck a bus terminal, killing nine people and injuring roughly 60 others.

Kenyan police have blamed the strikes on Somalia's Al-Qaeda-linked Shebab Islamists or their sympathisers, claiming the attacks are a response to the Kenyan army's incursing into Somalia, launched in October.

Nairobi sent troops into neighbouring Somalia following a spate of kidnappings in Kenya, which it blamed on the Islamists.

The operation aimed to curb the Shebab's influence in Somalia, where they control much of the south and central region in a country that has lacked a stable government for two decades.