The militant group al-Shabab said on Twitter that an American unmanned aircraft had come down near a town under its control.
A regional governor told Reuters that fighters had shot at the object.
The US uses drones in Somalia to support the Somali government and African Union (AU) forces as they battle the al-Qaeda-linked al-Shabab.
The US Africa Command has not confirmed the reports.
‘They hit it’
Residents in the town of Bulo Marer told the BBC they had seen the wreckage of a small plane but did not know whether it was a US drone.
Lower Shabelle region governor Abdikadir Mohamed Nur told Reuters news agency that Islamist militants had shot at the aircraft over the town for several hours before it came down.
“Finally they hit it and the drone crashed,” he said.
Al-Shabab said in a Twitter post that photographs of the drone would be published in the coming hours.
A French soldier was among several people who died during a failed operation to free a hostage in the town of Bulo Marer in January.
The hostage, Denis Allex, kidnapped in Somalia in July 2009, was also believed to have been killed by captors during the operation.
Al-Shabab emerged as the radical youth wing of Somalia’s now-defunct Union of Islamic Courts in 2006.
It is fighting to create an Islamic state in Somalia.
The AU has about 18,000 troops in the country to help the government battle the insurgents.
Al-Shabab was forced out of Mogadishu in August 2011 following an offensive by AU and government troops.
But it still controls most villages and rural areas of southern and central Somalia.