The talks came amid news that Central Intelligence Agency director Mike Pompeo had recently travelled in secret to North Korea to meet with Mr Kim ahead of a US-North Korea summit planned in the next two months.
About whether Japan will be exempted from new USA restrictions on imports of steel and aluminum, Kudlow said, "It's a key point on the agenda" of the summit meeting. "We'll iron those out, hopefully".
But Trump later suggested there was one area where he and Abe would have to agree to disagree: the Trans-Pacific trade partnership, which Trump pulled the USA out of days after his inauguration, but has recently said he might be open to re-joining.
As the two men sat for dinner Tuesday, it was revealed that Mike Pompeo, Trump's Central Intelligence Agency director and pick for secretary of state, had traveled to North Korea for preliminary talks with Kim early this month.
Trump, who was hosting Abe and his wife at his Mar-a-Lago resort in Florida, said a campaign of maximum pressure on North Korea would continue until Pyongyang agreed to give up its nuclear weapons.
Trump could take solace in the fact that none of the reporters asked the two world leaders who they were wearing.
Japan has also been questioning why it wasn't granted exemptions to Trump's protectionist measures on steel and aluminum when most other key US allies - among them Australia, Canada, the European Union and Mexico - have been. The meeting will be the third inter-Korean summit since the Koreas' 1945 division.
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It could get another boost after labour-market data showed the quickening of wage increases. "Earnings data will be key too". Wage growth has accelerated to a near three year high amid hopes a long squeeze on wages is finally coming to an end.
Mr Trump said he was working to reduce that imbalance and pushing to remove barriers to USA exports. Talks on Wednesday focused on trade, including the possibility of exempting Japan from new United States tariffs on steel and aluminum imports.
Nevertheless, Pompeo's conversations in North Korea had fueled Trump's belief that productive negotiations were possible, according to a US senior official briefed on the trip.
Several Senate Democrats, along with Republican Sen.
Donald Trump has once again changed his position on the Trans-Pacific Partnership trade deal days after just days ago raising the prospect he might consider rejoining the free trade agreement.
Kim is expected to meet South Korea´s President Moon Jae-in next week for landmark talks at which discussion of a formal peace declaration is now on the cards.
Melania Trump sat next to Akie Abie at dinner and is expected to give the 55-year-old a tour of the historic mansion President Trump acquired in 1985. The two remain united, he said, in their goal to have a complete and verifiable denuclearization of the Korean peninsula.