“He’s been weak, he’s been ineffective,” Republican candidate Mr Trump said of Mr Obama in a Fox News interview on Tuesday.
He mocked Mr Obama’s handling of the crisis with Russia over Ukraine.
Earlier, Mr Obama said Mr Trump was unfit to be president, and questioned why his party still backed him.
“There has to come a point at which you say: ‘Enough’,” Mr Obama said.
French President Francois Hollande joined the chorus of criticism on Tuesday, saying that Mr Trump made people “feel nauseous”.
He warned that a Trump presidential election victory could herald a very strong turn to the right around the world.
‘Look at Ukraine’
Speaking to Fox, Mr Trump said Mr Obama had been “the worst president, maybe, in the history of our country”.
Mr Trump has also been condemned for backing the Russian annexation of Crimea.
But he retorted on Tuesday: “I believe I know far more about foreign policy than he [Mr Obama] knows.
“Look at Ukraine. He talks about Ukraine [and] how tough he is with Russia. In the meantime they took over Crimea.”
Mr Obama and Mrs Clinton, his one-time secretary of state, had “destabilised the Middle East” while putting the “country at risk” with Mrs Clinton’s use of a private email server, he said.
Elsewhere on Tuesday, Mr Trump turned on two senior figures in his own party who have publicly criticised him.
In an interview for the Washington Post, he refused to endorse House Speaker Paul Ryan and Senator John McCain, who are up for re-election in November.
‘Unfit to serve’
Mr Trump is under fire for attacking the parents of a dead US Muslim soldier after they criticised him at the Democratic convention last week, when Hillary Clinton accepted her party’s nomination to fight him at the election on 8 November.
At the convention, Khizr Khan – a Muslim whose son was killed serving in the US military in Iraq – criticised Mr Trump’s plan to temporarily ban Muslims from entering the US.
Mr Trump responded by attacking the couple – who are called in the US a “Gold Star” family, the term for families that have lost a close relative in war. Democratic and Republican leaders as well as veterans’ groups quickly condemned him.
“The Republican nominee is unfit to serve as president and he keeps on proving it,” Mr Obama said on Tuesday.
“The notion that he would attack a Gold Star family that made such extraordinary sacrifices… means that he is woefully unprepared to do this job.”
New York Representative Richard Hanna became the first Republican member of Congress to publicly say he would vote for Mrs Clinton.
Mr Hanna said Mr Trump’s comments about the Khan family had been the deciding factor.
Until recently, many Republicans opposed to Mr Trump had stopped short of supporting Mrs Clinton, saying they would vote for a third party or “write-in” candidate.