The row between David Meikle, top, and George Redmond, above, took place in the Ivory Hotel, left, in Glasgow's Queen's Park area Main photograph: Martin Shields
David Meikle has alleged that George Redmond, who also chairs Strathclyde Partnership for Transport (SPT), challenged him to a fight in a hotel and called him a "prick".
But Redmond said Meikle triggered the row by telling him to "f*** off", and denied threatening him.
The bad blood between the pair can be traced back to a council debate on February 21 on the UK Government's "bedroom tax".
Meikle, the only Conservative in the city chambers, defended a policy its critics say will hammer the poor. Redmond responded by telling Meikle that "your family will be affected by this, your extended family will be affected by this", which angered the Tory councillor.
The tensions escalated two days later at the Ivory Hotel in the city's Queen's Park area.
Meikle and a friend, Andrew Morrison, had gone to the hotel after a leafleting round. As the pair sat in the hotel, Redmond and his wife also entered.
According to Meikle, Redmond approached his table. Meikle said: "Initially he wanted to apologise for his remarks at the council meeting - but did so in a patronising way, trying to touch my head.
"Having made clear my view that I did not accept it [the apology], Redmond became very aggressive - he challenged me to a fight outside, he gesticulated towards me, used offensive language [and] called me a 'prick'. He said, right in my face, using a twisting hand motion, he would 'smash me'."
Morrison said Redmond had offered Meikle a "half-hearted apology" that was "intimidating", which he said led Meikle to swear at the Labour member.
Morrison added: "He said something like, 'I'll smash you,' or 'I'll destroy you'." Asked whether Redmond had threatened Meikle, Morrison said: "Yes. If David had taken up the challenge, there would have been a punch-up in the car park."
Two days after the acrimonious exchange, Meikle reported the matter to Strathclyde Police.
However, Redmond denied threatening Meikle and said the Tory councillor had caused the row. He said: "I offered him [Meikle] an apology and he told me to 'f*** off' - I walked away from him and he said something else. I was then very firm with him."
Redmond said there were five "independent and impartial" witnesses who could back up his version of events, adding: "It was a storm in a teacup and a reflection of David's immaturity. I never challenged him to a fight."
A spokesperson for Strathclyde Police said inquiries were ongoing.
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