Obama was speaking during a news conference with the prime minister of Singapore, who is visiting the White House.
The president is challenging Republican leaders to withdraw their endorsements of Trump.
Obama says Trump’s criticism of a fallen Muslim-American soldier’s family is the latest evidence that the GOP presidential nominee is unfit to lead America.
“There has to come a point at which you say: ‘Enough’,” Mr Obama said.
At last week’s Democratic National 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 “Gold Star” family, the term for families that have lost a relative in war. Democratic and Republican leaders as well as veterans’ groups quickly condemned him.
Mr Trump struck back by questioning whether Ghazala Khan had been allowed to speak. She said she is still too grief-stricken by her son’s death.
He has also suggested that he fears the general election “is going to be rigged” – an unprecedented assertion by a presidential candidate.
Trump’s extraordinary claim – one he did not back up with any immediate evidence – would, if it became more than just an offhand comment, seem to threaten the tradition of peacefully contested elections and challenge the very essence of a fair democratic process.
“I’m afraid the election is going to be rigged, I have to be honest,” the Republican nominee told a town hall crowd in Columbus, Ohio. He added that he has been hearing “more and more” that the election may not be contested fairly, though he did not elaborate further.
Trump made the claim after first suggesting that the Democrats had fixed their primary system so Hillary Clinton could defeat Bernie Sanders. Trump has previously backed up that thought by pointing to hacked e-mails from the national party that appeared to indicate a preference for Clinton. Still, the former secretary of state received 3.7 million more votes than Sanders nationwide and had established a clear lead in delegates by 1 March.
The celebrity businessman – who has been known to dabble in conspiracy theories, including claims that President Barack Obama was not born in the US and, more recently, that Senator Ted Cruz’s father was an associate of President John F Kennedy’s assassin – also claimed that the Republican nomination would have been stolen from him had he not won by significant margins.
He then asserted that November’s general election may not be on the up-and-up.
Mr Trump repeated the charge on Fox News, saying: “November 8th, we’d better be careful, because that election is going to be rigged. And I hope the Republicans are watching closely or it’s going to be taken away from us.”
Requests to Trump’s campaign for additional explanation were not returned.
The statement could be an effort by Trump to lay the groundwork of an excuse if he loses the general election. But if he were to be defeated in November and then publicly declare that the election results were bogus, his claim could yield unpredictable reactions from his supporters and fellow Republicans.
The Clinton campaign declined to comment about Trump’s remarks.
It became a frequent catchphrase of his during a low-water mark of his primary campaign this spring, when forces allied with Cruz managed to pack state delegations with supporters of the Texas senator. Trump also asserted that the Republican Party had changed the delegate allocation in the Florida primary to favour a native candidate, like Jeb Bush or Marco Rubio, at Trump’s expense.
In recent weeks, in an effort to woo angry Sanders supporters to his campaign, Trump has made the claim that the Democrats’ process was also rigged. Trump said Sanders “made a deal with the devil,” and said of Clinton: “She’s the devil.”