"There is a military option to destroy North Korea's (missile) program and North Korea itself," Graham said on NBC's "Today" show. "If there's going to be a war to stop them, it will be over there. If thousands die, they're going to die over there, they're not going to die here and (President Donald Trump) told me that to my face."
He continued: "I'm saying (military options are) inevitable if North Korea continues."
But, he added, "You can stop North Korea militarily or diplomatically. I prefer the diplomatic approach."
On Friday, North Korea launched the longest range
intercontinental ballistic missile it has ever tested -- and the second ICBM this month.
North Korean leader Kim Jong Un touted the test as a success and claimed the "whole US mainland" is now within range of his missiles.
The latest test shows that Pyongyang's missile program may be more advanced than previously thought.
Trump condemned the missile launch and said the United States would act to ensure its security.
He told reporters on Monday at his second full Cabinet meeting as President that his administration will be able to take care of North Korea but offered no specifics about what he plans to do.
"We will handle North Korea. We are gonna be able to handle them. It will be handled. We handle everything," Trump said after a reporter asked him about his strategy.
Graham said in the interview that Trump told him that there will be a war with North Korea over its missile program if it continues to threaten the US.
"You're making the President pick between regional stability and homeland stability," Graham said on "Today." "There will be a war with North Korea over the missile program if they continue to try to hit America with an ICBM. (Trump has) told me that. I believe him."
Vice President Mike Pence took a more measured response in an interview with Fox News that was released Tuesday.
"All options are on the table," he told Fox News.
"Nuclear weapons in the hands of a rogue regime in North Korea represent a threat to nations in the region, and now we know a threat to the United States of America."
Pence reiterated his belief that the "era of strategic patience is over" regarding the rogue nation, saying the US is "specifically calling on China to do more."
Asked about Graham's comments, Sen. Dianne Feinstein said the South Carolina Republican should get a classified briefing.
"My reaction is that Lindsey Graham should get a classified briefing like the ones I have had," the California Democrat told MSNBC's Andrea Mitchell. "It is all classified. But we know much more about these weapons and where they are. And what the difficulties are. That's all I can say."
Last week, a US official told CNN that the US government doesn't believe North Korea will be able to launch a reliable nuclear-capable intercontinental ballistic missile until early 2018.
The official told CNN that while North Korea can currently get a missile "off the ground," there are a lot of undetermined variables about guidance, re-entry and the ability to hit a specific target.
This story has been updated.