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Lower Shabelle region governor Abdikadir Mohamed Nur said that al Shabaab militants had shot at the aircraft over the town of Bulamareer for several hours before it crashed.
"Finally they hit it and the drone crashed," Nur told Reuters.
The insurgents confirmed that a drone had crashed but did not say if they had downed it.
"A U.S. drone has just crashed near one of the towns under the administration of the Mujahideen in the Lower Shabelle region," al Shabaab said on a social media account.
Although the United States does not report its activities in Somalia, drones have been used in recent years to kill Somali and foreign al Shabaab fighters.
Western nations are worried that Somalia will sink back into chaos and provide a launchpad for Islamist militancy despite a fragile recovery after two decades of war.
Last year, the Wall Street Journal reported that the Pentagon was seeking to send drones to Kenya as part of a $40 million-plus military aid package to help four African countries fighting al Qaeda and al Shabaab militants.
Bulamareer residents said al Shabaab fighters had kept them away from the crash site.
"Al Shabaab fighters surrounded the scene. We are not allowed to go near it," resident Aden Farah told Reuters.
Al Shabaab, which is affiliated with al Qaeda, said in January 2011 that a missile launched from a drone had killed Bilal el Berjawi, a Lebanese al Shabaab fighter who held a British passport.
Another missile killed four foreign militants south of the Somali capital Mogadishu in February 2012.
Al Shabaab were driven out of Mogadishu in late 2011 and are struggling to hold on to territory elsewhere in the face of attacks by Kenyan, Ethiopian and African Union forces trying to prevent Islamist militancy spreading out from Somalia.