Papa’s in Baxtergate opened earlier this year into a five-storey building that had been vacated by the NatWest bank.
The family chain, which also has sites on Scarborough's Foreshore Road, and Cleethorpes, is now looking to add to its offering by applying for an alcohol licence.
An application has been lodged with Scarborough Council by Soteris Papacaoullas to secure permission.
Papa’s wants to be able to serve alcohol to its customers seven days a week from 10am until 10pm.
The application is now out to consultation.
The decision by Papa’s to move into the popular seaside town proved a controversial one when it first came to light.
In total, 486 people wrote to Scarborough Council to object to the plans when they were submitted, many saying that there were enough places to buy fish and chips in Whitby already.
Others just begged for something new in their town.
The plans were approved by council planning officers, however, as the objections had nothing to do with planning law and, as such, could not be considered valid.