The city centre business said it reserves the right to refuse admission to those in tracksuits -- the ban is currently in place after 7pm on weekends.
However Nash's pub, opposite The Clock, said it hadn't considered introducing a code.
Barman Dougie Parminter said: "There is no dress code here. We don't even have doormen." Mr Parminter said that there was increasing numbers of patrons wearing tracksuits and runners, but they wouldn't be calling the shots on what people could wear.
"Generally everyone is very well-behaved," he said.
A representative for The Clock was unavailable for comment.
The move against casual wear comes shortly after pyjama- wearing was banned in some public spaces, shopping centres and supermarkets in Dublin.
A Dublin youth group was forced to ban members wearing pyjamas during the summer. A sign stated: "Call us old-fashioned, but we think young women should wear pyjamas when they go to bed, not in a youth club where there's young men and adult men."
At the time, agency boss and former model Celia Holman Lee said that she thought the practice was "disgraceful".
"I have no idea where this trend came from, the only thing I can think of is because supermarkets are opening now all night, that maybe people just started to do their shopping in their pyjamas."
In the UK, supermarket chain Tesco put up notices stating that "footwear must be worn at all times and no nightwear is permitted".