Things got so heated that Duncan, who owns Bannatynes gyms, threatened to get his staff to report Jeff - who has two sons with the late Big Brother star Jade Goody - to social services.
The row began after Jeff tweeted businessman Duncan with two issues he claimed to have encountered while visiting the gym.
He said: "Morning @DuncanBannatyne I have a two complaints to make against your gym in Thurrock. Misled on membership fee & unclear on child policies.
"Get kids up & out at 6.30am so I can train before work-refused entry to gym because of U'16 policy. That's fair but why was it ok yesterday?"
Duncan asked for Jeff's membership details before asking him: "I'm confused. You want to send under 16s to a gym alone?" Jeff then responded: "No, I want them to sit quietly on the sofa whilst I work out like they were allowed to the day before #consistency."
This, however, appeared to rile multi-millionare Duncan, who shot back: "If you abandon your children in our lounge again I will instruct my staff to phone social services."
That what isn't true? @DuncanBannatyne I complain on two very reasonable counts in a decent manner & you speak to me about social services?!— Jeff Brazier (@JeffBrazier) December 19, 2014
Jeff - who has raised his sons, Bobby and Freddy, singlehandedly since Jade's death from cervical cancer in 2009 - was furious at Duncan's response and pointed out that his son Bobby was "never out of his sight".
"Poor from you @DuncanBannatyne allowing all to think I left my 11-year-old child unsupervised at 8am in reception. #Misleading," he said.
"He was never out of my sight @DuncanBannatyne How malicious of you to suggest to people that he was. Again, YOUR staff allowed him entry, which was sadly incorrect @DuncanBannatyne.
"You've questioned my parenting instead of responding responsibly to a paying customers concern. I complain on two very reasonable counts in a decent manner & you speak to me about social services?!"
Not backing down, Duncan replied: "Anyway you have been told the rules very clearly by me now so you cannot deny that." He then added: "Well I am bored now so an going to bed. Goodnight."
After reading Reveal's article, Duncan reiterated his stance that "gyms are dangerous".
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