Moon, a progressive politician, told former South Korean first lady Lee Hee-ho during a phone call Monday, "President Trump should be conferred the Nobel Peace Prize, we [South Koreans] only need to bring the peace."
Lee, who visited Pyongyang in 2000 with her husband, former President Kim Dae-jung, had called Moon to congratulate him on the summit.
Lee told Moon he had "done a great job" and that the South Korean president should be honored with the Nobel Peace Prize.
Moon has been carefully avoiding suggestions from supporters who favor putting his name before the Nobel Peace Prize Committee.
Kim Dae-jung was awarded the 2000 Nobel Peace Prize following the June 2000 summit with Kim Jong Un's father, Kim Jong Il.
Kim Dae-jung, a lifelong dissident, was later criticized following revelations of a $500 million payment to the North to secure the historic meeting.
Presidential Blue House spokesman Kim Eui-kyom has said a coalition of lawyers and Moon supporters advancing the cause of a Nobel for the South Korean leader has "no relation" to Moon and the movement "is not desirable."
Trump may not be averse to the idea of a Nobel, and his administration and South Korea have credited his policy of "maximum pressure" against North Korea for Kim Jong Un's change in attitude.
CNN reported Trump supporters began to chant "Nobel" during a rally in Michigan on Saturday, where he said a North Korea meeting could take place in the next three to four weeks.
"I just want to get the job done," Trump said.