The message was sent as part of Trump's efforts to maintain good relations with Kim's grandson and current North Korean leader, Kim Jong-un, as they try to negotiate the dismantlement of the regime's nuclear weapons program, Bolton said Wednesday in an interview with US broadcaster PBS.
Monday marked the founder's 107th birthday, a major national holiday in the North.
"I don't know how President Trump can be more forthcoming in his efforts to have a good relationship with Kim Jong-un," Bolton said. "He sends him pictures. He sends him letters. Happy birthday to Kim Jong-un's grandfather on his birthday on April the 15th."
"Really, this has been a full-court press by the president," Bolton added. "And we will wait to see what Kim Jong-un does."
The hawkish adviser is widely believed to have been behind Trump's decision to walk away from his second summit with Kim in Vietnam in February without any deal.
The breakdown was attributed to differences over the scope of North Korea's denuclearization and sanctions relief from the US
Asked what steps the US needs to see from the North, Bolton said, "We need to see the strategic decision and the conduct necessary to eliminate North Korea's nuclear and ballistic missile programs."
The subject was discussed extensively during a summit between South Korean President Moon Jae-in and Trump in Washington last week, he noted.
"It's something that we're committed to. We have seen North Korea not come forward and accept what the president has called the big deal. But that possibility remains out there. And it's why the president is still willing to have a third summit with Kim Jong-un," he added.
The big deal calls for North Korea to give up all of its weapons of mass destruction programs in exchange for sanctions relief.
In Hanoi, Kim made a counterproposal to dismantle only the North's main nuclear facility in Yongbyon, leading to the summit's collapse. (Yonhap)