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South Korea welcomed the prospect on Sunday, with the Unification Ministry saying it expected to swiftly engage in talks with Pyongyang, while urging the need to restore a hotline link between the two.
Kim's comment came after the North urged the United States and South Korea last week to abandon what it called their hostile policy and double standards towards it, if formal talks are to be held on ending the 1950-53 Korean War.
North Korea's pursuit of nuclear weapons has complicated the question of a formal end to the war, which halted with an armistice, rather than a peace treaty, leaving US-led UN forces technically still at war with the North.
Constructive discussions offer a chance for solutions on issues such as "the re-establishment of the north-south joint liaison office and the north-south summit, to say nothing of the timely declaration of the significant termination of the war", Kim said.
Speaking on Tuesday to the UN General Assembly, South Korean President Moon Jae-in had repeated a call for a formal end to the war, but later said time was running out for such progress before his term ends in May.
In Saturday's remarks, Kim said she noted with interest the intense discussion in the South over the renewed prospect of a formal declaration.
"I felt that the atmosphere of the South Korean public desiring to recover the inter-Korean relations from a deadlock and achieve peaceful stability as soon as possible is irresistibly strong," she said. "We, too, have the same desire."
On Sunday, responding to the remarks, Seoul's unification ministry said in a statement, "For these discussions, the inter-Korean communication line must first be restored swiftly, as smooth and stable communication is important."
The hotline, maintained by South Korea's military to handle relations with Pyongyang, has not operated since August, as North Korea stopped answering calls.
But that possibility, and the momentum the leaders generated over three meetings, came to nothing.
North Korea has rejected US overtures on dialogue and the head of the UN atomic watchdog said this week its nuclear program was going "full steam ahead".