Washington (CNN) Former President Jimmy Carter says President Donald Trump has a shot at winning the Nobel Peace Prize if he's able to work out a significant peace deal on the Korean Peninsula.

"If President Trump is successful in getting a peace treaty that's acceptable to both sides with North Korea, I think he certainly ought to be considered for the Nobel Peace Prize," Carter, a Nobel Prize winner himself, said on Politico's "Off Message" podcast . "I think it would be a worthy and a momentous accomplishment that no previous president has been able to realize."

Even though Trump has not formally secured the peace treaty, House Republicans formally nominated him for the prize this month.

Asked in early May about whether he deserved the honor, Trump said with a grin, "Everyone thinks so, but I would never say it."

Trump is scheduled to meet with Kim on June 12 in Singapore and US officials are continuing to prepare for the summit, but administration officials have grown increasingly skeptical that the summit will come to fruition amid North Korean threats last week to pull out because of US military exercises.

Carter has worked with several presidents since leaving the White House to serve as a diplomat in negotiations with North Korea. He told the podcast that North Korea's volatility is expected.

"If they're under constant belief that the United States wants to attack them, even using nuclear weapons -- which many Democrats and Republican leaders in our country have mentioned as a possibility -- and that we are destroying their economy, and they know that they're starving to death primarily because the United States withholds food aid, for instance, just giving them surplus food that we can't ever use, then I can understand how they feel."

The former president added that he thinks "the next mediator, next negotiator -- maybe President Trump, I hope -- will reassure (the North Koreans) that (the US is) willing to give up some of those things -- the threat of attack on them -- and to lift the embargo."

"That would be a cheap price, in my opinion, to pay for a cessation of their nuclear program," Carter added.