First by Twitter, and then at a press conference in the White House Rose Garden, Trump said he likes the idea of going to the southern side of the demarcation line that separates the Koreas, where South Korean President Moon Jae-in met Kim on Friday.
"We're not there yet, but if this happens, President Trump deserves the Nobel Peace Prize", he said.
Moon and North Korean leader Kim Jong Un on Friday pledged to end hostilities between the two countries and work towards the "complete denuclearization" of the Korean peninsula in the first inter-Korean summit in more than a decade.
"President Trump can take the Nobel prize".
He has previously said the USA president "deserves big credit for bringing about the inter-Korean talks", which were the first between North and South Korea for more than a decade.
Trump's embrace of the site could carry some risk given that a "celebration" during his summit with Kim could be premature.
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Russian President Vladimir Putin and his South Korean counterpart Moon Jae-in discussed the Korea summit over the phone, the Kremlin said on Sunday.
His comments were the most specific to date about the logistics of the meeting, which Trump said has the potential to be celebratory in tone.
South Korean leaders at the time opposed the idea of a truce that left the peninsula divided, and were not signatories to the armistice, which was officially signed by the commander of North Korea's army, the American commander of the U.N. Command and the commander of the "Chinese People's volunteers", who were not officially claimed by Beijing at the time.
Asked if Moon proposed the site during his phone conversation with Trump on the weekend, the official said Panmunjom was "naturally mentioned" as they talked about an appropriate location.
Among the possible venues cited in the media are Switzerland, Singapore, Mongolia, Sweden and the USA territory of Guam.
"Personally I think it will be a success, we will see".
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Foreign Affairs Minister Julie Bishop appeared to support the push for a Trump peace prize Monday when she told ABC radio today the US President deserved to be honoured if helped broker peace on the Korean peninsula.
Perhaps in response to these fears, the leaders of USA -allied Japan and South Korea have sought to forge closer ties with the unpredictable Trump, and have also massaged his ego by crediting him for easing tensions with North Korea.
"Ultimately, they need a country where both leaders have the security they need, to have a country where they can meet in common ground.it's a small number of countries to be honest" said Lee. "If it's not a success, I will respectfully leave".
But many USA officials are doubtful that Mr Kim will actually agree to give up nuclear weapons after working so long to develop them.
South Korea also said Monday that it will remove propaganda-broadcasting loudspeakers from the border with North Korea this week as the rivals move to follow through with their leaders' summit declaration that produced reconciliation steps without a breakthrough in the nuclear standoff.
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