Quoting unnamed sources in Washington, South Korea's Chosun Ilbo newspaper said the gifts reflected Trump's belief that Kim would follow through on commitments the leaders agreed at their summit in Singapore last month.
Speaking to reporters aboard Air Force One as he flew to Montana, Trump said: "I really believe that he sees a different future for the North Koreans".
"I'm confident what he intended there was, 'We did reduce the threat, ' " Pompeo said last week at a Senate Appropriations subcommittee hearing.
Trump traveled to Singapore for a summit with Kim, and despite receiving no verifiable assurances that that North Korea will denuclearize, he stood side by side with Kim, American and North Korean flags intertwined, giving Kim the legitimacy he and his regime had long sought.
One hoped-for breakthrough during his visit would be the return of the remains of US GIs killed during the 1950-53 Korean War.
According to a "source in Washington", the song had been a previous topic of conversation between Trump and Kim.
Both issues are considered essential tests of whether Mr Kim is serious about negotiations. Morell said Pompeo is testing whether Kim is lying about his commitment "to work toward complete denuclearization".
"When are they going to hand us that piece of paper?"
Mr Pompeo is therefore under pressure to produce evidence Pyongyang is taking steps towards ending its programme.More news: Phoney war over: Apple and Samsung end their long-running patent dispute
"This is your third visit to our country so I guess you are used to it now", Kim Yong Chol told Pompeo at the guesthouse, according to the report.
"So far principal results of Singapore summit have been to sow doubts about United States commitment to its South Korean ally, to weaken sanctions on NK, and to raise questions about U.S. seriousness given enormous gap between rhetoric of denuclearization and reality", tweeted Richard Haas, president of the Council on Foreign Relations and a top foreign policy adviser to both Presidents Bush. "The more you come, the more trust we can build between one another".
Over the past quarter-century, North Korea has frustrated or outfoxed USA administrations that have attempted to stop and reverse its weapons development through diplomacy and sanctions.
USA policy has not changed and the State Department is "committed to a denuclearized North Korea", spokeswoman Heather Nauert told journalists accompanying Pompeo on the trip.
A day after the summit, Trump announced that the USA was canceling joint military exercises with South Korea that the North has traditionally called practice for an invasion.
South Korea, Japan and the US military appear to have been caught off-guard by some of Trump's tweets and comments since the June 12 Singapore summit with Kim. The State Department said its policy remains unchanged.
Pompeo's talks, which resume on Saturday, will be closely watched in the region.
At the world team championships in May, North and South Korea were due to face each other in the women's quarter-finals - but they were given permission to form a joint team instead, losing to Japan in the semi-finals.