Somalia's Al-Qaeda-linked Shebab militants on Saturday announced the appointment of a successor to their former leader who was killed in a US air strike.

The group also vowed to avenge the death of Ahmed Abdi Godane and said they would continue their fight to topple the country's internationally-backed government.

The statement from the group, posted on jihadi forums and verified with Shebab officials, came after Somalia's government warned of a wave of retaliatory attacks.

The Horn of Africa nation's president also offered Shebab fighters a chance to lay down their arms and seize on a 45-day amnesty, telling them government troops and the African Union's AMISOM force were on the brink of overrunning their territory.

The Shebab statement said they had named Ahmad Umar Abu Ubaidah as their new leader, described by Shebab sources as a close lieutenant to Godane -- although the name is seen as likely to be a pseudonym.

"Avenging the death of our scholars and leaders is a binding obligation on our shoulders that we will never relinquish nor forget no matter how long it takes," the Shebab statement said.

"By the permission of Allah, you will surely taste the bitter consequences of your actions," it added, while also renewing a pledge of allegiance to Al-Qaeda chief Ayman al-Zawahiri, Osama bin Laden's successor.

Throughout the week, the rebels had refused to confirm or deny reports of Godane's death.

On Friday the Pentagon confirmed Godane died in an attack in which US drones and manned aircraft rained Hellfire missiles and laser-guided bombs on a gathering of Shebab commanders.

US officials did not specify how his death was confirmed. But in similar cases in the past, US intelligence agencies have tested DNA samples and used information gleaned from eavesdropping.

Somalia's national security minister said the country was on high alert.

"Security agencies have obtained information indicating that Al-Shebab is now planning to carry out desperate attacks against medical facilities, education centres and other government facilities," Kalif Ahmed Ereg told reporters.

"The security forces are ready to counter their attacks and we call on people to help the security forces in standing against violent acts," he said, adding nevertheless that "we congratulate the Somali people" on Godane's death.

- Amnesty offer -

Godane, who was 37, had been fighting to overthrow the Somali government, carrying out a wave of suicide bombings, brazen commando attacks, assassinations and kidnappings.

He had also overseen the group's transformation from local insurgency to major regional guerrilla threat, widening the group's reach with attacks in countries that contribute to AMISOM.

He claimed responsibility for the July 2010 bombings in the Ugandan capital Kampala that killed 74 people, and the group also claimed the September 2013 massacre in the Kenyan capital's Westgate mall, a four-day seige in which at least 67 people were killed.

Reacting to Godane's death, Kenyan President Uhuru Kenyatta offered his "heartfelt thanks" to the United States for "finally allowing us to begin our healing process". He said the operation had provided "a small measure of closure" for victims of the Westgate attack.

Somali President Hassan Sheikh Mohamud said the death was "a chance for members of Al-Shebab to embrace peace."

"While an extreme hardcore may fight over the leadership of Al-Shebab, this is a chance for the majority of members to change course and reject Godane's decision to make them the pawns of an international terror campaign," he said.

He said the government was "willing to offer amnesty to Al-Shebab members who reject violence and renounce their links to Al-Shebab and Al-Qaeda - but for the next 45 days only."

"Those who choose to remain know their fate. Al-Shebab is collapsing," he asserted.

The strike against Godane came days after African Union troops and Somali government forces launched "Operation Indian Ocean", a major offensive aimed at seizing key ports from the Shebab and cutting off one of their key sources of revenue: multi-million dollar exports of charcoal.

AU forces are targeting Shebab on several fronts, with Ugandan troops leading the offensives against the main port of Barawe, south of Mogadishu. Ugandan army spokesman Paddy Ankunda also told AFP that his forces had given "the intelligence that enabled the decisive targeting" of Godane.

In Mogadishu, a city struggling to return to normality after decades of civil war, residents said they feared the group may just find a new figurehead and carry on as usual.

"It is like Osama bin Laden, whose death never ended the existence of Al-Qaeda, they will still continue violence and could even become worse," said Ahmed Moalim Duale, a Somali police officer.