Mr Tybring-Gjedde and Mr Per-Willy Amundsen are members of the anti-immigrant Progress Party.
The Nobels in economics, literature, physics and medicine are awarded in Sweden.
A group of USA lawmakers also are backing Trump's nomination for the 2019 Nobel Peace Prize.
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.
Four US presidents - Theodore Roosevelt, Woodrow Wilson, Jimmy Carter, Barack Obama - have so far received the most prestigious peace prize in the world.
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In response to one reporter's question, Trump also confirmed that one name not on his list for potential pardons was O.J. Trump said that he wasn't thinking about pardoning either man, as neither has yet been convicted of a crime.
Despite the fact that the Norwegian Nobel Committee is appointed by the Parliament, its decisions are independent. Two Norwegian lawmakers have nominated Trump for the Nobel Peace Prize.
Norwegian Progress Party leaders nominated the USA president a day after the summit ended. The meeting came after a relatively diplomatic year for North Korea.
Nominations for this year's prize also formally closed on February 1, before the summit was agreed.
The committee's secretary, Olav Njoelstad, said "it is not impossible" for someone who has been criticised by committee members to be considered, and even win, the prize. The former has likened the hijab to a Nazi uniform and demanded stricter immigration controls.
Currently, there are "330 candidates for the Nobel Peace Prize for 2018 out of which 216 are individuals and 114 are organizations".
Mr Amundsen said: "A process is under way to ensure world peace in the future".