"It may be the first time the Nobel Peace Prize was given and there was mass casualties because I think a lot of liberals would kill themselves if they did that", Graham said on "Sunday Morning Futures With Maria Bartiromo". "Where if things work out, there's a great celebration to be had on the site".
A former South Korean President, Kim Dae-jung, won the prize in 2000 for his role in setting up a previous summit with North Korea, and his widow suggested Monday that Moon should also get the award.
Such talk reflects rising hopes for coming discussions between Trump and Kim, the first summit meeting between the leaders of the United States and North Korea, which could happen by early June.
The time change comes after the North and South Korean leaders met on Friday in an effort to formally end the 1950-53 Korean War.
Many other countries want the summit because "it has the chance to be a big event", Trump said.
Bolton said President Trump is "ready to go" on with meeting with Kim Jong-un, but the date and location for the summit have not been "pinned down yet".
In March he told Express.co.uk: "I think it will most likely happen in May".More news: Nirmal, Kavinder Swapping Status As BJP Finalises Its Cabinet Reshuffle
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Trump said the North Korean decision to close the test site, along with its suspensions of nuclear and ballistic missile tests, were hopeful signs.
The Trump administration said its goal in the upcoming negotiations was "the complete, verifiable and irreversible denuclearisation of North Korea".
As recently as January of this year, Trump took to Twitter to threaten nuclear engagement against Kim and North Korea, just months after using his favorite social media site to label Kim "Rocket Man".
Those details, including such hard issues as ensuring compliance through inspections and the potential lifting of economic sanctions on the North, would have to be worked out in subsequent meetings between the two countries, as well as South Korea, China, Japan and the United Nations. "If it's not a success, I will respectfully leave". Much of the agreement depends on whether Trump will agree to Kim's demands that the USA remove its nearly 30,000 military personnel from South Korea, as well as removing its "nuclear umbrella" security commitment to South Korea and Japan. They just had a very good meeting.
North Korea conducted a powerful underground nuclear test last September, and Mr. Kim ordered a series of missile launches throughout 2017.