Kim Jong-un says he and Donald Trump can exchange letters "asking about health" anytime they want and their relationship is "still excellent".
But the North Korean dictator has warned that relations between Pyongyang and Washington will remain "hostile" until crippling sanctions are lifted and the US and South Korea end war games near the border.
Kim said the breakdown in talks with the US has raised the risk of reviving tensions on the Korean peninsula and he will only meet with Trump a third time if America comes with the right attitude.
He said his personal relationship with Trump is far better than the countries' relations, but he warned that he will not give in to US demands and he is prepared for the standoff to continue.
The pair once traded insults and threats, with Trump calling Kim "little rocket man" and the North Korean leader calling the US president a "mentally deranged dotard".
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Kim offered insight into their relationship in a speech to the Supreme People's Assembly on Friday.
According to state media, he said: "As President Trump keeps saying, the personal ties between me and him are not hostile like the relations between the two countries and we still maintain excellent relations, as to be able to exchange letters asking about health [or of greetings] anytime if we want.
"We will wait for a bold decision from the US with patience till the end of this year but I think it will definitely be difficult to get such a good opportunity as the previous summit."
The leaders have met previously in Singapore in June last year and Hanoi last February without a major breakthrough.
Washington is aiming to halt North Korea's nuclear weapons programme, while Pyongyang wants economic sanctions to be lifted and joint military drills by the US and South Korea to stop.
North Korea sees the drills as a rehearsal for invasion.
Kim had previously promised not to resume missile or nuclear tests.
But satellite images show workers have been active at missile and nuclear test sites as recently as last month, according to analysts.
Kim said he is open to meeting Trump a third time if the US changes its stance on sanctions and shows "true willingness" to improve relations, according to remarks reported by the North Korean state news agency KCNA.
He said he will wait until "the end of this year" for the US to decide to be more flexible.
Trump met with South Korean President Jae-in in Washington on Thursday, and expressed a willingness for another summit with Kim.
However, he said the sanctions would remain in place.
Kim accused the US of "further escalating the hostility to us with each passing day".
He said the US policy of sanctions and pressure is "as foolish and dangerous an act as trying to put out fire with oil".
But he said he would not hesitate to sign an agreement if it takes into account both countries' considerations.
After the Hanoi summit ended abruptly without a deal, Trump told reporters he and Kim "like each other" and they could become "good friends".
He added: "He's quite a guy and quite a character. I think our relationship is very strong."
The pair have come a long way from recent hostilities where they traded insults and threats of military attacks.
Trump once taunted Kim by calling him "little rocket man" and said threats against the US would be “met with fire and fury like the world has never seen".
Kim once vowed to "tame the mentally deranged US dotard with fire".
