"We've got this silly battle going on about ANWR," said Thompson, who voted in favor of drilling the refuge as a former Tennessee senator.
"We've got the reserves up there that can be tapped," he told a crowd of about 50 people packed into a small café in this early voting state. "They're not going to solve our problem, but it's one step we can take."
Thompson's remarks came a day after environmental and Native Alaskan groups asked a federal appeals court to block Royal Dutch Shell PLC's plans for exploratory oil drilling near the refuge. The U.S. Mineral Management Service decided this year to allow the energy giant to drill up to 12 exploratory oil wells in the Beaufort Sea off the northern coast of Alaska.
Thompson argued the United States can become less reliant on oil-based energy and pursue nuclear and clean coal development. The country must reduce its dependence on fuel from unstable partners in the Middle East, Russia and Venezuela, he said.
"So much of it is coming from problem areas in the world, and it's making us a hostage, in many respects," Thompson said in response to a question from the crowd. "What we've got to do is not be so dependent on the wrong places in the world."
Thompson also said maintaining a strong military could help global stability, which would keep oil prices consistent.
"That does more to keep oil prices under control, probably, than anything else," he said.
Thompson planned to talk with voters at an Anderson restaurant later Wednesday before making an appearance in Lexington.