George Conway, a lawyer and persistent conservative critic of Mr Trump, tweeted on Monday images from the American Psychiatric Association's "Diagnostic and Statistical Manual of Mental Disorders".
He highlighted pages showing diagnostic criteria for "narcissistic personality disorder" and "antisocial personality disorder".
It followed a weekend in which Mr Trump lashed out at a number of perceived enemies, from late night comedy show Saturday Night Live to deceased Republican senator John McCain.
On Tuesday morning, Mr Trump hit back, sharing a tweet about Mr Conway by Brad Pascale, Mr Trump's 2020 campaign manager, and captioning it: "A total loser!"
Mr Pascale claimed "Mr Kellyanne Conway" had been turned down for a job in the administration and was "either fired/quit" at the Department of Justice.
"Now he hurts his wife because he is jealous of her success. POTUS doesn’t even know him!" he wrote.
In fact, according reporting by the New York Times in 2017, Mr Conway dropped out of the running to head the Justice Department's civil division, citing family considerations. Last year, he said he did not take a post because he considered the Trump administration "a s***show in a dumpster fire".
Within minutes of Mr Trump's tweet, Mr Conway responded by reposting diagnostic criteria for narcissistic personality disorder and writing: "Tell us, @realDonaldTrump—which of these diagnostic criteria do you not satisfy?"
"Congratulations!" he added. "You just guaranteed that millions of more people are going to learn about narcissistic personality disorder and malignant narcissism! Great job!"
Last week, Mr Conway tweeted: "Whether or not impeachment is in order, a serious inquiry needs to be made about this man's condition of mind."
He later added: "His condition is getting worse."
On Monday, he said Americans should be "thinking seriously" about Mr Trump's "mental condition and psychological state, including and especially the media, Congress – and the Vice President and Cabinet".
Ms Conway, one of Mr Trump's highest-ranking advisors, later dismissed her husband's concerns, telling reporters at the White House, "I don't share those concerns".
"I have four kids and I was getting them out of the house this morning to talk to the president about substance, so I may not be up to speed on all of them," Kellyanne Conway said, referring to her husband's tweets.