When President Trump mused about brokering peace between North and South Korea during a recent campaign rally, the chants from his supporters rumbled through the Michigan sports complex: “Nobel. Nobel.”

For Mr. Trump, who has toggled with unsettling ease between volatile threats of “fire and fury” and impassioned calls for peace, the possibility of receiving a Nobel Peace Prize once seemed far-fetched. (The committee that awards the prize said this year that a nomination for him had been forged twice, by an unknown perpetrator whose motives remain a mystery.)

But the idea of his 2019 nomination, formally submitted by a group of 18 House Republicans and heartily endorsed by President Moon Jae-in of South Korea, has started to take root among his supporters over the past few weeks as his own potentially historic summit meeting with Kim Jong-un of North Korea looms.

“Everyone thinks so, but I would never say it,” Mr. Trump said with a laugh on Wednesday when asked if he deserved the prize. “The prize I want is victory for the world.”