The report came after Trump's claim on Friday, February 15, that Abe had nominated him for the prize.
Intriguingly, Asahi's report says Abe was prompted to nominate Trump after receiving an "informal" request from the US government.
Citing unnamed Japanese government sources, the United States' request for a Peace Prize nomination came after Trump met with North Korean leader Kim Jong Un in Singapore in June of past year.
According to The Guardian, Abe was asked about the story in a parliamentary session, and he responded by issuing a non-denial. "I am not saying it's not true", he told an opposition lawmaker, adding that the Nobel committee doesn't reveal nominations and he would refrain from commenting. I'll probably never get it.
N. Korea at historic turning point: State media
And I sat in those handsome chairs and we talked. "I have a very good relationship with Kim Jong Un ", Trump said ahead of his meeting.
Trump is planning to have a second summit with North Korean leader Kim Jong Un in Hanoi Feb. 27-28 after an unprecedented meeting previous year.
Trump claimed on Friday that Mr Abe put his name forward for the award.
"You know why? Because he had rocket ships and he had missiles flying over Japan", Trump said.
Kim Eui-kyeom, Moon's spokesman, said Moon believed Trump "has sufficient qualifications to win the Nobel Peace Prize" for his work toward peace between North and South Korea, which have yet to sign a peace treaty after their 1950-53 war.
The U.S. president didn't elaborate on when Abe sent the letter or showed him the copy.
Polish officials may cancel Israel visit amid Holocaust tensions
He later told Army Radio on Monday that he was in favor of maintaining relations with Poland but repeated his earlier statements. The EU condemns Israeli communities in Judea and Samaria and has said Israel uses excessive force against Palestinians in Gaza.
The Nobel Foundation's website says a nomination for the Nobel Peace Prize may be submitted by any person who meets the nomination criteria, which includes current heads of states. I'll probably never get it.
The deadline each year for nominations is midnight, January 31.
For over a year, both Koreas and the USA have worked toward a policy of rapprochement, which culminated in the June meeting between Trump and North Korean leader Kim Jong Un.
The Japanese leader's efforts have included many summit meetings and teleconferences with the USA president, pledges to buy expensive weapon systems produced by US firms and even a red-carpet reception for Ivanka Trump when she visited Japan in November 2017.
This year there are 304 candidates for the Nobel Peace Prize, out of which 219 are individuals and 85 are organizations, the Nobel Committee has said.
Train 18 ready for commercial run on Sunday: Railways
The Vande Bharat Express will run between New Delhi and Varanasi, stopping only at Kanpur and Prayagraj on the way. Meanwhile, the new Vande Bharat Express could make the same trip in less than eight hours.
Abe personally has a large political stake in making progress in resolving the abduction issue with North Korea, an important one for his nationalist political base.