During a press conference in Tokyo on Monday, Mr Trump said he agreed with North Korea when it called Mr Biden a “low IQ individual”.
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The Biden campaign responded that the president’s comments were “beneath the dignity of the office”.
Mr Biden is the current front-runner to face Mr Trump in next year’s election.
Defending his comments in a tweet on Tuesday, the US president appeared to suggest that he had been generous when quoting the North Korean leader Kim Jong-un’s description of Mr Biden.
“[Mr Kim] called him a ‘low IQ idiot’,” Mr Trump wrote, adding that he was “actually sticking up” for Mr Biden with a less offensive description of him.
During his four-day visit to Japan, Mr Trump disregarded the old Cold War axiom that US politics stop at the water’s edge.
The Republican president said on Twitter that he had “smiled” when Mr Kim called Mr Biden “a low IQ individual, & worse”.
Throughout the trip, Mr Trump continued to direct jibes at the former US vice-president, whom he calls “Sleepy Joe”.
Mr Biden’s campaign manager Kate Bedingfield said in a statement on Tuesday: “To be on foreign soil, on Memorial Day, and to side repeatedly with a murderous dictator against a fellow American and former vice president speaks for itself.”
Appearing on NBC’s Meet the Press on Sunday, White House Press Secretary Sarah Sanders defended the president, saying that Mr Trump and Mr Kim “agree in their assessment of” Mr Biden.
She added: “Again, the president’s focus in this process is the relationship he has and making sure we continue on the path towards denuclearisation.”
Mr Trump has frequently criticised Mr Biden since the former vice-president announced his candidacy, predicting that he may become the eventual Democratic White House nominee.
Mr Biden’s public schedule has been relatively light over the past two weeks following its launch last month.