"We are assessing the results of the operation and will provide additional information as and when appropriate," Pentagon spokesman John F Kirby said in a statement.
Abdikadir Mohamed Nur, governor of Lower Shabelle Region, who is with the advancing forces, told the BBC that the attack was successful.
"US drones managed to hit the representative of al-Qaeda in Somalia, who is also the leader of al-Shabab, Ahmed Godane," he told the BBC Somali service.
"We can tell that a senior figure from the group was killed due to the way they reacted after the attack, as they have started committing atrocities in the area, they have beheaded some of the people who had mobile phones and arrested many others [accused of spying]."
Residents of Hawai village, some 240km south of the capital, Mogadishu, say they heard three loud explosions and then saw thick black smoke rising.
One resident told the BBC he had seen the burnt-out remains of a car and three trucks but al-Shabab fighters prevented him from seeing if there were any bodies inside.
The US Rewards for Justice programme issued this photograph of him in 2012
- Seen as al-Shabab's chief ideologue and spiritual leader
- Comes from Somalia's northern breakaway region of Somaliland
- Also known as Mukhtar Abu Zubair
- Studied in Sudan and Pakistan, where he became radicalised
- Said to have fought in Afghanistan
- Reputed to be a good orator and poet
- Helped form al-Shabab after a union of Islamist courts briefly took power in Mogadishu in 2006
- Sentenced to death in absentia for an attack in 2008 in Somaliland's capital, Hargeisa
- Became al-Shabab's top commander after US air strike killed his predecessor Aden Hashi Ayro in same year
- Pledged allegiance to al-Qaeda in 2009
- US put $7m (£4m) bounty on his head in 2012
The 22,000-strong African Union force says it has taken some towns from al-Shabab in recent days.
The American action comes after al-Shabab rebels attacked a detention centre in Mogadishu on Sunday, in an apparent effort to free other militants detained there.
Al-Shabab militants on Sunday unsuccessfully tried to free other extremists held in a Mogadishu detention centre
Somali officials said all of the attackers, as well as three government soldiers and two civilians, were killed.
The Pentagon and the US state department have supported the AU force that has driven al-Shabab from their former strongholds in the capital and some other urban centres since 2011.
The militants continue to carry out bombings and assassinations in Mogadishu.
Al-Shabab fighters want to overthrow the internationally supported Somali government and frequently attack government targets as well as neighbouring countries that provide troops to the AU force.
The group said it carried out last year's attack on the Westgate mall in the Kenyan capital, Nairobi, in which at least 67 people were killed.
