Sam Smith's, which is based in Tadcaster and runs around 200 premises across the country, told The Times earlier this month that they were not taking contact details from customers because they believed it breached GDPR rules.
Government guidance says pubs, cafes and restaurants should keep a record of customer phone numbers so they can be told if they have come into close contact with anyone who has tested positive for Covid.
But it is not a legal requirement.
The issue was raised at a health scrutiny meeting in Wakefield on Thursday afternoon.
Councillor Martyn Johnson said he'd become aware of the issue at The Wheel Inn, on Bradford Road in the village of Wrenthorpe, which is run by the brewery.
He told the meeting: "We have three pubs and a club in Wrenthorpe. One is a Sam Smith's pub and the owner has told his landlords and pub managers that they're not to take any details or contact numbers for test and trace.
"That concerns me because no-one can enforce it and it concerns me because of the spread of the virus."
Stephen Turnbull, from Wakefield's public health team, said the council would consider lobbying government to make test and trace a legal requirement.
He added: "It isn't enforceable. What we're trying to do is to encourage establishments to use test and trace and to keep a record of names.
"It can be really helpful if we do have an outbreak linked to a bar or a club. It can spread very quickly.
"I totally agree, it's a concern."
Sam Smith's has been contacted for comment but is yet to respond.
In response to The Times article on August 18 a company spokeswoman said: "The reasoning behind [not having track and trace] is it’s against GDPR data protection to ask people’s names and addresses and most people would give false names and addresses.
"Sam Smith’s customers are locals and most managers know the customers and word would get around if Covid was in a pub.”
"People who write down names and addresses with a pen on paper could also spread the virus. There is also confidentiality — there was a man who followed a pretty woman into a pub and saw her write down her name and phone number and then copied it and bothered her."
Mr Turnbull said test and trace had been used effectively following an outbreak at Truth bar, in Wakefield city centre, earlier this month.
Eight members of staff at the premises tested positive for coronavirus.
Mr Turnbull said the bar was effectively blameless for the outbreak and that it had been running safely.
He revealed that one customer who was tracked down had tested positive, but it was believed they had contracted the virus from elsewhere.
Local Democracy Reporting Service