Mike Graham accused Labour MP Lisa Nandy of not doing her job "for three years" during a heated interview.
The spat began after the politician accused the talkRADIO host of trying to "mislead the public" by referring to a deal during their discussion, rather than a Withdrawal Agreement.
"She [Theresa May] negotiated a Withdrawal Agreement which settled legal and technical questions, and allowed the UK then to go and negotiate a deal. Those negotiations haven't started yet," Ms Nandy said.
After the Wigan MP referred to Mike as "difficult", he hit back: "I do my job every day, you guys haven't done your job for three years."
But the politician called his comments "complete nonsense".
"You are not doing your job now," she continued. "You are pretending to the public that all we're doing in Parliament is running around, trying to stop a deal."
In response, Mike insisted that "that's what it looks like".
"I talk to the public every single day and that is what they say to me," he added.
"They are sick to death of politicians arguing over semantics, arguing over little niceties and making excuses about why we cannot do what they asked you to do, which is leave the European Union."
Discussing the exchange online, one Twitter user wrote: "I have a lot of time for Lisa Nandy despite never voting Labour in my life, but this was a car crash. Completely incomprehensible."
Another posted: "She's guilty of fallacy of equivocation over 'deal'. Was deliberately obfuscating meaning of 'deal' & used it in a different sense to MG [Mike Graham]. Here, 'deal' means deal we currently have with EU, the WA & the trade deal yet to be decided. She used all three meanings interchangeably."