'This is not a reality show,' Obama tells Trump

US President Barack Obama warned on Friday that occupying the Oval Office "is not a reality show", in a swipe at outspoken Republican candidate Donald Trump who is vying to replace him in the White House.

Fighting with Obama is a battle Trump would likely relish as he tries to rally support within his own party. During hard-fought Republican primary campaigns, the billionaire delighted in responding to attacks from rivals and found that his support grew when he lashed out at his opponents.

Asked about Trump at a media briefing in the White House, Obama called on the press and public to weigh past statements by the Republican but did not point to any specific issues or remarks.

"This is not entertainment," Obama said, a reference to Trump's television background. "This is not a reality show. This is a contest for the presidency of the United States," he said.

Some top Republican leaders - US House Speaker Paul Ryan among them - are still expressing wariness about Trump, who became the party's presumptive nominee this week when two Republican rivals dropped out of the White House race.

Former Florida Governor Jeb Bush, who ran unsuccessfully for the Republican nomination, posted on Facebook that he will not vote for Trump - one of the sharpest slights yet against the New York real estate mogul by a senior Republican.

"Donald Trump has not demonstrated that temperament or strength of character. He has not displayed a respect for the Constitution. And, he is not a consistent conservative. These are all reasons why I cannot support his candidacy," Bush wrote, adding that he would not vote for likely Democratic nominee Hillary Clinton either.

US Senator Lindsey Graham from South Carolina, who made an unsuccessful bid for president, joined a growing list of Republicans who are refusing to support Trump and he announced on Friday he will also skip the Republican convention in July. Mitt Romney, who won the Republican nomination in 2012, is also refusing to support Trump.

But Trump on Friday won the endorsement of another former Republican presidential nominee, Bob Dole, who lost to Bill Clinton in 1996.

For Trump, finding unifying enemies like Obama and Hillary Clinton could help rally Republicans back to his side ahead of the November 8 general election.

Obama is likely to be the feature of much of Trump's criticism in the general election. Republicans have sought to paint Clinton as an extension of the Obama administration who would continue all of his policies.