"All security measures and procedures have been taken to deal with the situation, and procedures for gathering evidence and investigation are being completed," the police said in a statement on Tuesday.
The United States embassy in the capital Muscat said it was following reports of the shooting incident in Wadi al-Kabir, in the governorate of the capital city Muscat.
"US citizens should remain vigilant, monitor local news and heed directions of local authorities," it said in a statement.
Oman has maintained its neutrality in the turbulent region and mediated in disputes including between the United States and Iran.