Donald Trump has dismissed Barack Obama's time in the White House as a "disaster" after the US president said he was not fit to succeed him.
"He's been weak, he's been ineffective," Republican candidate Mr Trump said of Mr Obama in a Fox News interview on Tuesday.
Mr Obama has questioned why Mr Trump's party hasn't disowned him.
Mr Trump has also 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.
Republican donor backs Clinton
Amid the feuding within Republican ranks, prominent party donor and fundraiser Meg Whitman has publicly endorsed Democrat Hillary Clinton, saying Donald Trump's "demagoguery" had undermined the national fabric.
"To vote Republican out of party loyalty alone would be to endorse a candidacy that I believe has exploited anger, grievance, xenophobia and racial division," she wrote on Facebook.
"Trump's unsteady hand would endanger our prosperity and national security. His authoritarian character could threaten much more."
In other developments:
- A Reuters/Ipsos opinion poll suggested Hillary Clinton had extended her lead over Mr Trump to eight percentage points, from six points on Friday
- A federal judge who has been a target of Mr Trump's repeated scorn denied a media request to release videos of the candidate testifying in a lawsuit about the now-defunct Trump University; Mr Trump's lawyers had argued the videos would have been used to tarnish his campaign.
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 his comments that appeared to back the Russian annexation of Crimea.
But he retorted: "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.
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.
At the convention, Khizr Khan, whose son died while serving 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.
Republicans not voting for Mr Trump
- Barbara Bush, former first lady
- Jeb Bush, former Florida governor, 2016 presidential candidate
- William Cohen, former secretary of defence
- Jeff Flake, Arizona senator
- Lindsey Graham, South Carolina senator, 2016 presidential candidate
- Larry Hogan, Maryland governor
- John Kasich, Ohio governor, 2016 presidential candidate
- Mark Kirk, Illinois senator
- Mitt Romney, former Massachusetts governor, 2012 Republican presidential nominee
- Ileana Ros-Lehtinen, Florida congresswoman
- Ben Sasse, Nebraska senator
Republicans voting for Mrs Clinton
- Richard Armitage, former deputy secretary of state
- Hank Paulson, former treasury secretary
- Brent Scowcroft, former national security adviser
- Richard Hanna, New York congressman
- Meg Whitman, party donor and fundraiser