By Joseph Akwiri

MOMBASA, Kenya (Reuters) - A prominent Kenyan Islamist, accused by the United States and U.N. Security Council of supporting the Somali militant group al Shabaab, was shot dead on Tuesday, police said.

Abubakar Shariff, also known as Makaburi, was killed as he left a court compound about 15 km (10 miles) north of the port city of Mombasa, police chief for Kisauni area Richard Ngtia told journalists. Makaburi had been attending a court hearing.

Makaburi and another man were outside the court waiting to be picked up when another vehicle approached and the men were sprayed with bullets, he said. Both were killed.

"Our brother Abubakar Shariff Makaburi has left us. He is dead," a preacher at a mosque in Kisauni, a Muslim-dominated area near Mombasa, said through a loudspeaker.

"May his soul rest in peace. He has died a brave death."

A Reuters witness saw Makaburi's corpse with bullet wounds to the body and head, before police took it away. Dozens of Makaburi's supporters gathered nearby demanding police hand over the corpse. Police fired in the air to disperse the crowd.

Makaburi's death could stir fresh unrest in the coastal area where most of Kenya's Muslims live. Muslim youths clashed with police for three days in February after a man was killed during a police raid on a mosque used by firebrand preachers.

Kenyan police have dismissed Islamist charges that they are to blame for extra-judicial killings. Two other leading Islamists have been killed in the past two years.

The east African country, the region's largest economy, is still reeling from an al Shabaab attack on a shopping mall in Nairobi in September in which at least 67 people were killed.

Kenya is trying to break up militant recruitment networks among its Muslim community in an effort to end attacks by Somali Islamist militants and sympathizers bent on punishing it for sending troops to Somalia to fight al Shabaab rebels.

"I KNOW I WILL BE KILLED"

Tensions on the coast have increased since an attack last month by gunmen on a church in the Mombasa area in which six worshippers were killed. There was no claim of responsibility, but it was similar to other such assaults blamed on Islamists.

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A report by the U.N. monitoring group on Somalia said Makaburi had an influential role in the Kenyan-based Islamist militant group al Hijra, suggesting he had called for attacks on a range of targets in Kenya.

The group said al Hijra was working on behalf of al Shabaab.

"He became the central figure in terms of al Hijra, in terms of the operations, in terms of relations with al Shabaab," one Western diplomat said of Makaburi.

He said Makaburi had become the most high-profile Islamist activist in Kenya after the killing of Islamist cleric Aboud Rogo in August 2012.

"I know I will be killed," Makaburi had told Reuters in October, saying the police would seek ways to justify shooting him. "I'm ready to die for it. If they want to or if they don't, they will give me martyrdom."

The U.N. Security Council and the United States had accused Makaburi of raising funds and recruiting for al Shabaab.

(Additional reporting by Drazen Jorgic in Nairobi; Writing by Edmund Blair; Editing by Andrew Roche)