After Jeff, 35, tweeted his confusion at Bobby, 11, and Freddie, 10, not being allowed in the gym with him, Duncan hit back.
Jeff wrote: "Get kids up & out at 6.30am so I can train before work – refused entry to gym because of U16 policy. That's fair but why was it OK yesterday?"
Duncan, owner of gym chain Bannatyne's, replied: "You abandoned your children in the gym. They are your responsibility, not my staff's.
"Look after them properly. If you abandon your children in our lounge again I will instruct my staff to phone social services."
But Jeff insisted Duncan's claims are 'malicious' lies. He said: "SO @DuncanBanntyne you forgot to say which sofa YOUR staff had allowed Bobby yo sit on, the one IN the gym directly where I trained!
"He was never out of my sight. How malicious of you to suggest to people that he was. Again, YOUR staff allowed him entry."
Other Twitter users responded to the spat, with some suggesting Duncan was being unprofessional by dealing with the matter online.
One said: "@DuncanBannatyne is supposed to be a businessman. Does he not find it embarrassing treating a customer like this online?"
Jeff has looked after Bobby and Freddie since their mum Jade Goody's death in 2009.