Earlier, North Korea's official KCNA news agency described Biden as a "rabid dog" that will soon pass away and needs to be put down, adding that he was allegedly exhibiting the "final stage of dementia".
"I am the only one who can get you where you have to be". Trump responded with a tweet Sunday morning, acknowledging that Biden is "actually somewhat better than that".
In the past, however, Biden has accused Trump of being too cozy with 'dictators and tyrants, ' and has been highly critical of his meetings with North Korean leader Kim Jong Un, for which there have been three.
North Korea had lambasted Mr Biden for having the "temerity to dare slander the dignity" of its leader, Mr Kim.
The president's comments were included in a retweet of a conservative commentator's post about North Korea's attack on Mr Biden. "They must be beaten to death with a stick". He also said that there would be "no love letters to a Biden administration".
'That's because he'd restore American leadership in the world on day one by putting our security, interests, and values at the heart of our foreign policy, ' he continued.
US Defense Secretary Mark Esper said on Sunday that joint military exercises with South Korea scheduled for December, were postponed to support diplomacy with North Korea. Trump and Kim Jong Un have in the past exchanged "beautiful" letters.
In response to the North Korean jibe, Mr Biden said he saw such insults "as a badge of honour".
Last week, the North's chief nuclear negotiator, Kim Myong-gil, claimed that Washington had suggested another round of working-level talks in December.
Talks between the two sides on a deal for North Korea's denuclearization stalled after a Hanoi summit between Kim and Trump broke up in February.