Democratic President Barack Obama has denounced Donald Trump for his proposed US ban on Muslim immigrants, joining Hillary Clinton in portraying the Republican presidential candidate as unfit for the White House.

Clearly annoyed, Obama responded to Trump's proposed suspension of immigration from countries with a "history of terrorism" after Sunday's killing of 49 people in an Orlando, Florida, nightclub.

Obama and Clinton, the Democrat he has endorsed to succeed him in the November 8 election, made nearly simultaneous speeches. The gunman was US-born Omar Mateen, 29, whose parents immigrated from Afghanistan 30 years ago.

Trump had criticised Obama for not using the term "radical Islamic terrorism" to describe Islamic State militants.

"What exactly would using this label accomplish, what exactly would it change?" Obama replied.

"Someone seriously thinks we don't know who we're fighting? ... There's no magic to the phrase 'radical Islam'. It's a political talking point. It is not a strategy.

"Calling a threat by a different name does not make it go away. This is a political distraction," said Obama, criticising the "yapping" and "loose talk" he said he hears from Republicans.

Obama, who cancelled a joint campaign appearance with Clinton planned on Wednesday in Wisconsin after the events in Orlando, appeared to be enjoying his role in the campaign to select his successor. He tangled with Trump in 2011, producing his birth certificate to refute Trump's claim that the president was not born in the US.

"We now have proposals from the presumptive Republican nominee for president of the United States to bar all Muslims from immigrating to America. We hear language that singles out immigrants and suggests entire religious communities are complicit in violence," Obama said. "Where does this stop?"

On Tuesday, Trump was unrelenting in his criticism of Obama, saying in a statement that Obama "claims to know our enemy, and yet he continues to prioritise our enemy over our allies, and for that matter, the American people.

"When I am president, it will always be America first," said Trump.

Aides said Trump, who on Monday said Obama should resign for failing to handle the threat properly, would have more to say at a rally later in Greensboro, North Carolina.

The crossfire overshadowed the last Democratic presidential primary on Tuesday in Washington DC.

Clinton, Obama's former secretary of state, addressed supporters in Pittsburgh.

The candidate said Trump's proposal bolstered her case that he was temperamentally unfit to serve as president, saying the commander in chief "is a job that demands a calm, collected and dignified response" to events like the Orlando massacre.

Clinton said Trump seemed to suggest on Monday in a television interview that Obama might have somehow been responsible for the deadliest mass shooting in modern US history, a point that Trump said he did not make.

"I have to ask: Will responsible Republican leaders stand up to their presumptive nominee or will they stand by his accusation about our president?" she said.

The biting criticism was likely to increase the discomfort among many establishment Republican leaders about Trump with little more than a month until party figures gather in Cleveland on July 18-21 to formally nominate him.

House Speaker Paul Ryan, the top US elected Republican who was his party's 2012 vice presidential nominee, on Tuesday distanced himself from the proposed Muslim ban in a further sign of establishment unease with Trump's agenda.

"I do not think a Muslim ban is in our country's interests," said Ryan, who last year criticised Trump's original proposal for a temporary ban on Muslims entering the US.

Ryan and like-minded leading Republicans have struggled to reconcile their desire to unify the party before a tough fight against Clinton while at the same time separating themselves from some of the positions and rhetoric of Trump, who defeated 16 rivals to win the presidential nomination battle.