Graham continued, "Donald Trump is not going to capitulate, so there’s really only two options -- peace or war."
The senator said he supports a call by several leading Democratic senators for the U.S. not to lift sanctions against North Korea without verifiable dismantling of its nuclear and missile arsenal. The Democratic senators set out their recommendations in a letter to Trump last week.
But Graham said Democrats need to help Trump in his talks with North Korea by supporting an authorization to use military force against the country if necessary.
"I appreciate you telling the president what a good deal would look like, but the country needs you to back the president up to get that deal," Graham said on "This Week," addressing the Democratic senators.
"So here’s the question for my Democratic colleagues," he added. "If diplomacy fails, will you support my efforts to authorize the use of military force as a last resort to convince North Korea and China things are going to be different this time?"
"A bipartisan [military-force authorization] would really make that letter [from Democrats] much more credible,” Graham said. “And if diplomacy fails, as a last resort, Democrats and Republicans need to put the military option on the table or we’ll never get a good deal."
While Graham said the U.S. needs to be prepared to take military action against North Korea if necessary, he doesn't expect that to happen soon.
"I don't expect a deal next Tuesday. I expect the process to be started next Tuesday,” the South Carolina senator said. “We’ll know diplomatic failure when we see it."
He added, "We’ll find out in about a year if [talking with North Korea] is going to work."
Stephanopoulos asked if one year was a realistic timeline, noting that a top U.S expert on North Korea's nuclear weapons said it will take 15 years to fully dismantle the program.
Graham said, "What I’m saying here is that you’ll have a deal one way or the other in [Trump's] first term that can be implemented in a way that we all believe ... I don’t think 15 years is on the table, but I don’t expect it to be done in one year."
"What I do expect to be done in a year from now is an agreement that does dismantle their nuclear weapons programs, their missiles, removes all plutonium and uranium," the senator said.
"We're not going to let [North Korea] run out the clock again," Graham said. "They talk about giving up [weapons], but they wind up building up ... They promise a bunch of things, then they back out. Trump is going to call the question on North Korea while he’s president of the United States."
A leading Democratic senator who appeared later on "This Week" Sunday was asked about whether he would support Graham's draft authorization to use military force against North Korea.
Sen. Bob Menendez, D-N.J., the ranking member on the Senate Foreign Relations Committee, told Stephanopoulos he's "not ready" to authorize military action.
“I'm not ready to give an authorization for the use of military force to this president or any other one until I understand that the path for peace is not attainable," Menendez said. He said he would also want to know whether "the threat continues to be a real challenge to the national security of the United States, and we have all the intelligence and robust debate in the Senate on such a [military-force authorization]."