In remarks at the Democratic National Convention, Mrs Obama offered a sharp rebuke of the Trump presidency, telling viewers he “has had more than enough time to prove that he can do the job, but he is clearly in over his head”.
“He cannot meet this moment,” she told the convention on Monday, held online due to the coronavirus outbreak.
Mr Trump retaliated in a series of Twitter posts in which he claimed Barack Obama was the reason he currently held the position of president.
“Somebody please explain to @MichelleObama that Donald J. Trump would not be here, in the beautiful White House, if it weren’t for the job done by your husband, Barack Obama,” said Mr Trump on Twitter.
During her speech Mrs Obama praised the Democratic presidential rival for the November election, Joe Biden, as a “profoundly decent man” who “knows what it takes to rescue an economy, beat back a pandemic and lead our country”.
But Mr Trump said Mr Biden was “merely an afterthought” on Twitter and slammed Mrs Obama’s praise as “very late & unenthusiastic endorsement”.
He added: “My Administration and I built the greatest economy in history, of any country, turned it off, saved millions of lives, and now am building an even greater economy than it was before. Jobs are flowing, NASDAQ is already at a record high, the rest to follow. Sit back & watch!
“The ObamaBiden Administration was the most corrupt in history, including the fact that they got caught SPYING ON MY CAMPAIGN, the biggest political scandal in the history of our Country. It’s called Treason, and more. Thanks for your very kind words Michelle!”
The US President claimed the Biden administration was “corrupt” as he critiqued Mr Obama’s handling of the swine flu outbreak.
“Looking back into history, the response by the ObamaBiden team to the H1N1 Swine Flu was considered a weak and pathetic one,” said Mr Trump in his series of Twitter posts.
“Check out the polling, it’s really bad. The big difference is that they got a free pass from the Corrupt Fake News Media!”
Mr Trump’s critics defended Mr Obama by claiming the current US President had “inherited” the economy from Mr Obama, and pointed out that the US currently has the highest tally of coronavirus deaths in the world.
Michelle Obama was joined by Senator Bernie Sanders as they headlined the first night of a two-hour-a-night virtual meet up for the conference which was broadcast across the world.
After she left the White House in early 2017, Mrs Obama has become an author and global celebrity.