The Los Angeles Times said the 266-foot rig is holding more than 150,000 gallons of fuel and other petroleum products.
The Los Angeles Times said the rig was being towed from Dutch Harbor last week when its main tow line and its emergency line detached during conditions of high winds.
Authorities said the towing vessel's engine then failed.
The Los Angeles Times said the crew of the Royal Dutch Shell rig was evacuated and authorities made the decision to drop the last tow line over the weekend during stormy weather.
The rig hit ground Monday night and has since been beached on an island in southern Alaska near Kodiak Island. Officials said two attempts to survey damage to the rig failed Tuesday as a result of high winds.
Officials with the Alaska Department of Environmental Conservation said emergency response teams are monitoring threats to water or wildlife, but have no indications of any so far.
The Times said the rig is near a habitat for endangered sea lions.