The convoys came under attack with small fire arms late on Sunday, said a US military spokesman.
These were the latest in a string of attacks on US forces and American targets in the Gulf state which was used as the launch pad for the US-led invasion of Iraq in March.
State-run Kuwait Television, quoting Kuwaiti security sources, said the US soldiers fired back at the attackers.
The assaults came a few hours after news that ousted Iraqi President Saddam Hussein was captured in Iraq, news welcomed by many Kuwaitis whose country was invaded by Saddam’s forces in 1990.
Thousands of US troops are based in Kuwait, which is also being used as a transit point for American soldiers going home on vacation, or returning to their bases in Iraq.