"Can you imagine if Don Jr had the problems of 'where's Hunter'?" Mr Trump, 74, told hundreds of supporters at a Make America Great Again event at Prescott Regional Airport.
"Don Jr. Oh, my poor boy. Boy, what that boy has been through," the president added before going on to praise his other children, Ivanka, who "floats through life" and Eric, who has "been great".
The president appeared to be suggesting that if his son Don Jr, 42, had drug problems "they wouldn't allow me to run". It was not immediately clear what Mr Trump was referring to or who "they" were.
Trailing Mr Biden by a wide margin in most major polls with just two weeks to go until election day, team Trump has ramped up its attacks on Hunter, 50, who has battled substance abuse.
Mr Biden has spoken candidly about his son’s struggles with alcohol and drugs, including in the recent TV election debate.
Last week, a story published in the right-leaning New York Post newspaper alleged evidence of corruption involving Hunter Biden’s business dealings in Ukraine.
The Post also published a photo of Hunter Biden lying in bed with a crack cocaine pipe in his mouth.
The story claimed an email allegedly found on a laptop at a Delaware hardware store showed the former vice president met with a Ukranian gas executive, using his influence to help his son.
Republicans have long sought to tie the former vice president to his son's business activities, with top Trump figures and the Post itself describing the supposed email as "smoking gun evidence".
"Can you imagine if Don Jr had problems of where's Hunter? Don Jr. Oh, my poor boy. Boy, what that boy has been through." -- Trump pic.twitter.com/gquLsPd6Rp
— Aaron Rupar (@atrupar) October 19, 2020
But Team Biden said the meeting described in email never took place. There have been several concerns about the veracity of the Post story, with one expert describing the email cache as a "Trump campaign product".
Trump's personal lawyer, Rudy Giuliani, is alleged to have handed the materials included in the story to the Post. The FBI is now reportedly investigating whether the alleged emails were obtained as part of a foreign intelligence operation.
During yesterday's rally - the third day of a western campaign swing - Mr Trump lashed out at Mr Biden, 77, for pledging to heed the advice of scientific experts, saying dismissively that his rival "wants to listen to Dr Fauci."
Mr Trump had earlier in the day publicly rebuked Dr Fauci after the infectious disease expert criticised him during a Sunday interview.
He also repeated attacks on the news media, singling out NBC's Kristen Welker, the moderator of the next presidential debate, as well as CNN for aggressively covering a pandemic that has so far claimed some 220,000 US lives.