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Republican U.S. presidential candidate Donald Trump speaks at a campaign rally in Eugene, Oregon Reuters/Jim Urquhart

Donald Trump and Senator Elizabeth Warren have spent a week trading insults on social media amid rumours she is being considered as a running mate by Democrat hopeful Hillary Clinton.

The Republican presumptive presidential nominee wrote eight tweets about the Massachusetts senator, describing her as "goofy" and a "phony Native American", while Warren hit out at Trump for "racism, sexism and xenophobia".

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The snipes started with Trump's attack after it was suggested Warren had made Democrat front-runner Clinton's vice president shortlist, on which the presidential hopeful confirmed there was at least one woman.

Trump tweeted last week: "I hope corrupt Hillary Clinton chooses goofy Elizabeth Warren as her running mate. I will defeat them both."

Warren has thus far refused to endorse Clinton or her Democrat rival Bernie Sanders, although she is more closely aligned with Sanders' policies and would also not be a surprise choice for Sanders' VP if he won the nomination – although this is unlikely given Clinton's lead on him of 2,240 delegates to his 1,473.

But since Warren's name has come up as a potential Democrat VP, neither she nor Trump have shied away from confronting one another on social media.

In response to Warren's criticism of Trump's comments about Hillary Clinton "playing the woman card", Trump wrote: "Goofy Elizabeth Warren is now using the woman's card like her friend crooked Hillary. See her dumb tweet 'when a woman stands up to you..."

Warren wrote a barrage of tweets aimed at Trump over the course of a day, in which she referred to the GOP candidate as "dangerous", writing: "Your policies are dangerous. Your words are reckless. Your record is embarrassing. And your free ride is over."

Despite the pair both dedicating time to insulting one another, with Trump insisting Warren didn't run for president due to her "phony Native American ancestry", Trump added: "Isn't it funny when a failed senator like goofy Elizabeth Warren can spend a whole day tweeting about Trump & gets nothing done in Senate?"

Not to be outdone, Warren shot back: "@realDonaldTrump can't talk about Wall St, college costs or min wage, so he spent all day belching insults. Pathetic. Really pathetic."

Trump's insults have seen him gather quite a social media following, but it isn't often he receives such a dressing down on Twitter from the opposing side.

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