"I believe we can accomplish this by sending a clear signal, namely by awarding Trump the Nobel Peace Prize".
Trump then said that "everyone thinks he deserves" the Nobel peace prize, but he wants another award - "victory for peace".
Some 216 individuals and 114 organisations have been nominated for the 2018 prize, which will be announced in October.
But before placing your bets, you may want to read what some members of the five-strong Norwegian Nobel Committee, which chooses the victor of the $1 million prize each October, have written about the USA president in the past.
And in a press conference after the document signing - the details of which were not released to Congress or US allies ahead of time - President Trump said he had agreed to terminate joint military exercises with South Korea, which have been going on since the 1950s.
Trump and North Korean leader Kim Jong Un signed an agreement on Tuesday pledging to denuclearize the Korean Peninsula, though it's unclear what the timeline is for denuclearization and how the USA will enforce it.More news: Justify: Racing's $75-Million Horse?
The nomination missed this years' deadline for the prize, so Mr Trump's nomination would only make him eligible for the award next year.
The Norwegian Nobel Committee's vice-chair, Henrik Syse, wrote in a blog in December, 2016, that: "President-elect Donald Trump has, with his rhetoric, lowered the threshold for moral decency so far it is downright scary".
The prizes are administered by the Norwegian Nobel Committee, appointed by Norway's parliament. Trump neglected to notify South Korea, or the USA military, of those plans before the press conference. A record 330 people were nominated this year.
Some Republicans have floated Trump's name for the Nobel Prize in recent months as he worked towards negotiations with North Korea.
The last American to win the prize was former President Barack Obama, who won in 2009.