A formal repatriation ceremony would be held at Osan Air Base on Wednesday, the statement said.
The first step toward advancing this longer process, which was the last item both leaders agreed to, is North Korea's return of the remains of the U.S. servicemen from the war.
"Today's actions represent a significant first step to recommence the repatriation of remains from North Korea and to resume field operations in North Korea to search for the estimated 5,300 Americans who have not yet returned home", it said.
Trump had previously said that the North had already returned the remains of about 200 service members. While the active period of the war lasted three years, warring parties never signed a peace treaty, so the Korean War is technically still ongoing.
Menendez, the ranking member of the committee, called Trump's meeting with Kim "a reality TV "summit" that was little more than a photo-op with a brutal dictator".
July 27 marks the 65 anniversary of the Armistice Agreement when the US, North Korea and China signed a ceasefire to halt three years of brutal fighting which claimed 4 million lives. They would then be flown to Hawaii after an initial DNA analysis at Osan air base in South Korea.
"The adoption of the declaration on the termination of war is the first and foremost process in the light of ending the extreme hostility and establishing new relations between the DPRK and the USA", the North's Korean Central News Agency said in a statement on Tuesday, referring to North Korea by its official name, the Democratic People's Republic of Korea.
North Korean and United Nations forces delegates meet at the cross-border village of Panmunjom to discuss an armistice on October 11, 1951.
The return of the remains is mainly symbolic, as over 5,300 U.S. personnel never returned home from the conflict.
CNN's Barbara Starr reported that each case of remains was to be opened and photographed inside North Korea, before take off.
He said Trump remained upbeat about the prospects for North Korean denuclearisation, but Kim needed to follow through on his summit commitments.
"The adoption of the declaration on the termination of war is the first and foremost process in the light of ending the extreme hostility and establishing new relations between the DPRK and the US", the North's Korean Central News Agency said in a statement on Tuesday, referring to North Korea by its official name, the Democratic People's Republic of Korea.
He said the United States was engaged in "patient diplomacy" to persuade North Korea to give up its nuclear weapons, but would not let the process "drag out to no end". However, reviving it could complicate USA efforts to persuade countries around the world to maintain economic pressure on Pyongyang over its ballistic and nuclear programs.