US Republican Party presidential contender Donald Trump has disclosed a rival candidate's mobile number in a fiery speech that further escalated his feuding with his own party.

Senator Lindsey Graham, whose phone number Trump read to the audience and who recently called the billionaire businessman a "jackass", saw his phone line jammed by floods of Trump supporters following the disclosure.

Trump is now at odds with much of the Republican establishment after a series of provocative comments, topped by his weekend mocking of Senator John McCain's experience as a tortured prisoner of war in Vietnam.

Since then the property developer and reality TV host has intensified his criticism of McCain, and his record on veterans issues in the Senate, even as politicians from both parties and veterans groups have rushed to his defence.

Trump, although doing well in recent polls, is still considered a long shot among the crowded field of candidates for the Republican Party nomination.

But in this early stage of the campaign, at least, the attention on Trump has drawn the spotlight away from presumed favourites such as former Florida Governor Jeb Bush.

In a speech to hundreds of supporters in South Carolina on Tuesday, Trump kept up the drumbeat, accusing McCain of being soft on illegal immigration.

"He's totally about open borders and all this stuff," Trump said.

McCain sparked Trump's temper last week when he said his inflammatory remarks about Mexican immigrants had brought out the "crazies".

McCain said on Tuesday he would no longer respond to Trump's comments.

'World's biggest jackass'

Republican leaders and other 2016 candidates have been frustrated by Trump's brash campaign, which has often overshadowed their own in recent weeks.

But many Trump backers in early-voting states argue the exchange is being overblown by the media and his opponents, who fear his sudden rise in early polls.

Even many non-Trump supporters doubted the latest controversy would hurt his appeal among a slice of the electorate that is deeply frustrated with the Washington establishment and finds his unvarnished persona refreshing.

Graham, a friend of McCain and one of the 16 notable Republicans running for the presidential nomination, betrayed the growing exasperation and anger of many in the party when he appeared earlier on "CBS This Morning."

"Don't be a jackass," Graham said. "Run for president. But don't be the world's biggest jackass."

He said Trump had "crossed the line with the American people" and predicted this would be "the beginning of the end with Donald Trump".

Trump responded during his speech by calling Graham an "idiot" and a "lightweight".

He then held up a piece of paper and read out the senator's mobile phone number to the capacity crowd of 540 people and the TV audience.

"Give it a shot," Trump encouraged. "He won't fix anything, but I think he'll talk to you."

Graham's voice mailbox was full Tuesday afternoon, his spokeswoman Brittany Brammell said. Graham tweeted later: "Probably getting a new phone. iPhone or Android?"

Source: Agencies