"I am always ready to sit down again with the U.S. president at any time and will make efforts to produce an outcome that the global community would welcome", Mr Kim said, in comments translated by the Yonhap news agency.
But a continuation of the current USA policy of sanctions and pressure may compel the North to "find a new way" to defend the country's interests and achieve peace and stability on the Korean Peninsula, he said.
One thing that was different from past speeches by the North Korean leader was that it was "a relaxed, indoor setting, with him sitting in a comfortable chair rather than standing and speaking in a huge square", the Korea Times said.
"If the United States responds to our active and pre-emptive efforts with trustworthy steps and corresponding behaviour, [relations] will move forward at an excellent and fast pace", Mr Kim said in his speech. Pyongyang over the past months has accused Washington of failing to take corresponding measures following the North's unilateral dismantlement of a nuclear testing ground and suspension of nuclear and long-range missile tests.
After months of bitter acrimony and public name-calling between the two leaders, Trump and Kim met in Singapore in June.
He also called for a halt to the deployment of "strategic assets" on the Korean Peninsula, an apparent reference to US bombers and submarines capable of carrying nuclear weapons.
Those measures were in line with its resolve to "no longer make, use or spread" nuclear weapons, Kim said, indicating a possible moratorium on weapons production for the first time.
"This whole diplomatic process is not the way the US does diplomacy, but it is the way Asia tends - is more comfortable doing diplomacy as well, this leader-driven, kind of piece by piece, small commitments at a time, to build this relationship and build a track record and build trust as you move towards a mutual goal", she said. While Pyongyang conducted no nuclear or missile tests a year ago, satellite images have pointed to continued activity at related facilities.
Culminating in late 2017, the North has carried out 6 atomic blasts and launched rockets capable of reaching the entire USA mainland, but has now carried out no such tests for more than a year.
There was no immediate comment from the White House.
Kim's speech also points toward a hard year for the U.S. It will be critical for the allies to develop a firmer policy as it's clear the issue of joint drills is coming to a head, Mount said.
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And it does not include the $5 billion Trump has demanded, instead allocating $1.3 billion for border security and fencing. To retaliate over the Democrats' refusal to fund the wall, Trump is refusing to sign a wider spending package.
Kim also met with Moon three times in 2018 - twice at the border truce village of Panmunjom and once in the North's capital - and at the weekend vowed to meet Moon "frequently" this year. Kim said an inter-Korean military agreement reached in their last summit in September to reduce conventional military threats was "realistically a non-aggression declaration".
As a way to tackle chronic power shortages, he raised the need for atomic energy.
North Korea seems determined in 2019 to receive some sort of sanctions relief - or what Kim referred to as "corresponding measures" - as part of any bargain with the Trump Administration."Duyeon Kim, an adjunct senior fellow at the Center for a New American Security: "As expected, he didn't rock the boat, clearly being mindful of the diplomatic process with Washington and didn't brag about his nuclear might".
Such is the mistrust that Mr Kim is desperate to meet Mr Trump again just to get some movement on these preliminary issues.
North Korea however would have "no option but to explore a new path in order to protect our sovereignty" if the United States "miscalculates our people's patience, forces something upon us and pursues sanctions and pressure without keeping a promise it made in front of the world", Kim said.
The U.S. has suspended large-scale military exercises with South Korea.
The hardening stalemate has fueled doubts on whether Kim will ever voluntarily relinquish the nuclear weapons and missiles he may see as his strongest guarantee of survival.
Little progress has been made since Mr Kim met US President Donald Trump in Singapore in June past year.
Seoul and Washington are in a security alliance and the United States stations 28,500 troops in the South.
The North Korean leader is asking Washington to take the next step to break the stalemate, such as allowing the reopening of the joint Kaesong Industrial Complex, analysts said.