It was unclear if there were any casualties.
The terror group has claimed responsibility for the attack, the report said.
The Associated Press reports the US uses the base to launch drone attacks on al-Shabab targets.
A Somali official says a suicide auto bomber detonated a vehicle packed with explosives at the gate of a military airstrip which is a base for US and Somali forces.
The second attack was by a suicide auto bomber targeting Italian peacekeepers in Mogadishu.
The bombs were followed by small arms fire, a security source told Reuters. The U.S. routinely conducts airstrikes against al-Shabab fighters in Somalia, launching about 50 so far this year.
The base is in the Lower Shabelle region, about 60 miles (100km) west of Mogadishu.
The Al-Shabaab militant group struck a United States airbase in Somalia and a convoy of European Union military advisers in separate attacks on Monday, a security officer and witnesses told AFP.
BBC- Jihadists have attacked a military base where USA soldiers train commandos in Somalia, causing casualties, reports say.
"After breaching the perimeters of the heavily fortified base, the mujahideen [holy warriors] stormed the military complex, engaging the crusaders in an intense firefight".
Al Shabaab is fighting to overthrow the weak, United Nations -backed Somali government and enforce its own strict version of Islamic law.