The as-yet-unnamed business would be home to 12 rescue cats who would live in specially constructed accommodation on the premises.
Customers would be able to interact with the felines in the main area of the cafe, which is a former interiors shop on Goodramgate in York city centre.
Kitty Cafe Leeds: A gallery of ridiculously cute photos of the catsThe unit will be extended to three floors, with the top storey being reserved for the cats and a separate kitchen area.
Capacity would be restricted to 22 customers to protect the cats' wellbeing.
Leeds already has a hugely successful cat cafe, Kitty Cafe, which opened in December 2017. It is home to more cats - around 35, which are all available for adoption - and also runs themed events in the evenings. Sheffield also has a cat cafe.
The prospective owners of the York cafe have argued that it will become a visitor attraction for tourists as well as residents.
One full-time and three part-time jobs would be created.
Three years ago, a crowdfunding campaign was set up to launch a cat cafe called The Scratching Post in York, but it failed to meet its target.
The proposals are now awaiting a decision from City of York Council planners.