The situation on the Korean Peninsula saw a significant improvement past year, with North Korean leader Kim Jong-un boosting ties with Seoul and holding a historic summit with Trump in Singapore in June.
Mr Trump's claim that Mr Abe had sent him a "beautiful copy" of a letter sent to the Nobel committee could not be immediately verified. "In line with this policy, I would like to refrain from giving a comment", Abe said in response to a question from an opposition lawmaker.
During the same press conference, Trump also claimed that former president Barack Obama told him he was thinking about "starting a big war with North Korea" and complained that Obama had received the prestigious prize after being in office "for about 15 seconds" and doing nothing to deserve it.
"Being Trump's closest friend among world leaders has not worked out too well for Abe", said Jeff Kingston, director of Asian Studies at Temple University Japan.
U.S. President Donald Trump listens to Japan's Prime Minister Shinzo Abe during a bilateral meeting on the sidelines of the 73rd session of the United Nations General Assembly in NY, U.S., September 26, 2018.
"Cooperating hand-in-hand with President Trump, I'd like to make all-out efforts to resolve the nuclear and missile issues as well as the abduction issue, as the most important agenda for our country", Abe said.
The US is Japan's ally and anchor for national defense and Abe has assiduously cultivated cordial ties with Trump.
Trump believed that his historic meeting with Kim was potentially a ticket for a Nobel Peace Prize that would become an essential part of his legacy, his advisors and White House officials told the Associated Press.
US President Donald Trump and North Korea's leader Kim Jong Un meet at the start of their summit at the Capella Hotel on the resort island of Sentosa, Singapore, on June 12, 2018.
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This season and last season, I was on the bench a lot of the time, and always I tried to score. "Son was fantastic again". Jadon Sancho: "I've been very impressed by him - he's a player with incredible qualities".
The five-strong Norwegian Nobel Committee, which decides who wins the award, does not comment on nominations, keeping secret for 50 years the names of nominators and unsuccessful nominees.
Abe was one of the first world leaders to embrace Trump after the 2016 presidential election.
"I am asking them to give you the Nobel Peace Prize"," Trump told reporters. "North Korea summit", the prime minister said. "I said "thank you".
The deadline each year for nominations is midnight on January 31.
"They gave it to Obama", Trump said.
The request from Washington followed the first U.S.
"He's not making Abe look very good".
There are 304 candidates for the Nobel Peace Prize for 2019 out of whom 219 are individuals and 85 are organizations. The two leaders are due to meet later this month in Hanoi, Vietnam.