The US president and Kim have held three meetings since June 2018 but negotiations over denuclearisation have been largely deadlocked since the breakup of a summit in Hanoi last February.
Before the collapse of that meeting, Trump had said that he and Kim had an excellent relationship -- and had even "fallen in love".
Kim Kye Gwan, North Korea's foreign ministry adviser, said on Saturday that Trump's congratulatory letter to Kim had arrived in the isolated state directly from the United States.
"As acknowledged by the world, it is true that the personal relations between (Kim Jong Un) and President Trump are not bad," he said in a statement carried by the official KCNA news agency.
However, it would be "absent-minded" to expect Pyongyang to resume dialogue because of that warm personal relationship, the statement continued.
On the gridlocked nuclear talks, he said that reopening dialogue would only be possible in the case of Washington's "absolute agreement" on the issues raised by North Korea in previous talks.
But the senior official was sceptical about the US accepting these demands, saying: "We know well that the US is neither ready nor able to do so. We know... the way we should go and will go on our way."
At a meeting of his ruling party in December, Kim Jong Un declared an end to North Korea's moratoriums on nuclear and intercontinental ballistic missile tests.
The self-imposed ban on such tests had been a centrepiece of the nuclear diplomacy between Pyongyang and Washington over the past two years, which has seen little tangible progress.
Saturday's statement on KCNA came a day after South Korea's presidential security adviser said Trump had asked Seoul to deliver his birthday message for Kim Jong Un.
North Korea has never officially confirmed Kim's age or date of birth -- but on January 8, 2014, basketball star Dennis Rodman sang him "Happy Birthday" before an exhibition match in Pyongyang.
Trump and Kim's personal relationship, while rocky, appears to have warmed significantly since their pre-2018 animosity that saw them exchange insults and threats of war.
However, Kim Kye Gwan accused Washington of trying to manipulate any warm feelings between the pair for political gain.
"We have been deceived by the US, being caught in the dialogue with it for over one year and a half, and that was the lost time for us," his statement said, adding that Kim Jong Un's "good personal feelings" about Trump would not influence any future talks.
"There will never be such negotiations as that in Vietnam, in which we proposed exchanging a core nuclear facility of the country for the lift of some UN sanctions," it added.
"There is no need for us to be present in such talks, in which there is only unilateral pressure, and we have no desire to barter something for other thing at the talks like traders."
The adviser also warned Seoul against trying to act as a mediator between the US and the North: "The South Korean authorities had better not dream a fabulous dream that we would return to the dialogue with thankful feelings for the birthday greetings like someone."