Influential lawmaker Lindsey Graham, a foreign policy hawk, told NBC's Today Show: "There is a military option: To destroy North Korea's program and North Korea itself."
Last week, North Korean leader Kim Jong-Un boasted that his country could now strike any target in the United States after carrying out its latest intercontinental ballistic missile test.
World powers have been trying to stifle Pyongyang's weapons program through United Nations-backed sanctions, but have failed to daunt the regime and Washington is growing frustrated.
Graham said that if diplomacy, and in particular pressure from the North's neighbor China, fails to halt the program then the United States will have no choice but to take devastating military action.
"They've kicked the can down the road for 20 years. There will be a war with North Korea over the missile program if they continue to try to hit America with an ICBM," he said, describing his discussions with Trump.
"He's told me that. I believe him. If I were China, I would believe him, too, and do something about it. You can stop North Korea, militarily or diplomatically.
"I prefer the diplomatic approach. But they will not be allowed to have a missile to hit America with a nuclear weapon on top."