The pair exchanged a series of heated Twitter messages after the Apprentice boss claimed the singer has no talent.
"Can you tell me who the hell this Tulisa bird is on X Factor, she's in the celeb section of The Sun and Mirror every day. What has she done?" Sugar tweeted.
He added: "This Tulisa talks nonsense on X Factor, what qualifications does she have to judge singing talent. Please advise where I am missing the point."
Tulisa hit back: "As someone that came from nothing to something maybe u should have a little more respect for people that have done the same.
"I have grafted from the age of 11 with my band NDUBZ, we wrote & produced all our music, sold 3 platinum albums, 3 sell out tours, the last being an arena tour & a number 1 debut solo single! I have worked my bloody ass off!
"So mr lord of judgement - stick to judging on The Apprentice & I'll stick to what I know MUSIC #itsaNOfromme."
She later wrote to her followers: "Soz bout that tweet peeps. But that t**t has been bad mouthing me since last year! Who watched XF? Some amazing talent this year right?"
He replied: "What do you know? That is the point. Don't insult me by comparing my humble beginnings with yours."
Sugar also insisted that he is "the first to recognise talent", adding: "She has none."