The row has seen Trump, the host of the US version of The Apprentice, tell Sugar, the host of the UK show, "drop to your knees, Sugar, and say thank you, Mr Trump" and claim that Sugar owes his career to him. Trump fired a fusillade of increasingly outrageous insults at Sugar, branding the Amstrad entrepreneur "a total loser" and "dopey".
In an increasingly ludicrous war of words, the mega-rich tycoons traded insults over who was wealthier in an exchange which attracted widespread condemnation from followers of both businessmen.
The row started when Sugar responded to claims by Trump about wind farms on twitter. Trump had initially tweeted: "How can George Osborne reduce UK debt while spending billions to subsidize Scotland's garbage wind turbines that are destroying the country? You're all wrong—check the facts! UK is massively subsidizing Scotland's wind turbines & the people don't want them."
Sugar replied: "I think you have that wrong. The Scottish don't want you!"
For Trump, this was a red rag to a bull. All hell broke loose, with Trump tweeting to Sugar: "Unlike yours, my financials are phenomenal. People don't know your real numbers and would not be impressed."
Sugar hit back, accusing Trump of piling up debts. He tweeted: "Trump no one can get your numbers. 'Smoke and Mirrors'. My numbers published at UK companies house. How much do u owe banks?"
This sparked a furious series of tweets from the American mogul, as he demanded of Sugar: "Why don't you tell the public what you're really worth – they would be very disappointed." He added: "I'm worth more than $8 billion, acknowledged, almost no debt-"
He then attacked the Amstrad founder's role on The Apprentice, saying: "Your net worth doesn't even qualify you to host the Apprentice. Keep making me money."
Sugar retaliated: "My net worth is real Donny. Not sure u have money, you talk a good game. The people want to know how much do you owe banks."
Trump brushed this off, instead criticising Sugar's failure to foresee the success of digital music devices including the iPod. Trump tweeted: "Nice call on predicting that the iPOD would be 'dead, finished, gone, kaput'. Great business foresight."
Sugar responded: "Do you know what a iPod is? That's my worst mistake, you've been near bankrupt in past and sent companies into Chapter 11 [bankruptcy protection]."
As the feud escalated still further, Trump returned to the original nub of the dispute – the pair's contrary views on wind farm technology – by tweeting: "The wind turbines are ruining the beauty & majesty of Scotland- But you wouldn't notice because you have no vision and you are a total loser."
He added: "You need the income from the show [The Apprentice] to keep going – hope it doesn't hurt."
Sugar branded Trump a "stupid man", saying: "Every one knows when I did the show I gave the fee to charity - You are a bully and not used to someone giving you hell. I don't think you could write a cheque for $100m. You are full of s***. And the American public know it."
Keeping on the wind farm issue, Trump also tweeted: "If you think ugly windmills are good for Scotland you are an even worse businessman than I thought ... and, in my opinion, should not be doing The Apprentice."
The pair continued their battle over who really runs The Apprentice. Trump tweeted to Sugar: "Keep working hard so I make plenty of $ with your show ...You have little persona, but The Apprentice concept is great and lucky for you!"
He went on: "Unlike you, I own The Apprentice. You were never successful enough ... to do The Apprentice but I approved you anyway. Without my show you'd be nothing!'
Although Sugar deleted the tweets, he is said to have replied: "You approved me! What a load of rubbish dream on. You don't own apprentice, Mark Burnett [creator of the show] does. Shut up and argue with Obama. Success is measured with what you have in business. I own all my real estate with no bank borrowing, how about you big shot?"
With all trace of dignity by now thrown to the wind, Trump called Sugar "dopey" and added: "Look in the mirror and thank the real Lord that Donald Trump exists. You are nothing!"
In a parting blow, Trump reiterated his self-proclaimed $8bn worth, adding: "You're worth peanuts ... without my show nobody would even know who you are."
The row got increasingly bizarre with Piers Morgan, the British journalist who now anchors a show on US TV channel CNN, joining in from the sidelines and backing Donald Trump with tweets like "exactly!", "BOOM" and "greatest tweet of all time ... hahahahahahahaha".
Trump's opposition to wind farms has been extremely vocal since he discovered that offshore turbines were planned for construction close to his new £750 million Aberdeenshire golf resort.
The businessman is bitterly opposed to the renewable energy source because he feels it will spoil the views at his new development.
By the looks of the last few days on Twitter, though, some could be forgiven for thinking Trump could generate enough hot air to power a wind farm all of his own.