The application by the budget supermarket at the site off Harrogate Road was approved in August, and investigative work started this month.
Aldi submitted fresh plans for a 1,638 sq metre store at the site, which has been earmarked for development for years, after councillors warned earlier this year that they would reject the first set of proposals put forward.
Planning officers recommended the scheme for approval at a meeting of Leeds North and East Plans Panel in June.
But members of the committee sent the application back to Leeds City Council’s chief planning officer for a decision, asking that the designs for a public open space be included. Approval was finally granted in August.
One of the conditions is that no building or demolition work is allowed before 8am or after 6pm Monday to Friday, after 2pm on Saturday or on Sundays and Bank Holidays.
Planners also ruled that no more than 20 per cent of the floor space shall be used for non-food shopping, to avoid adversely affecting other local shops.
Outside of the normal opening hours of the store as retail premises, the car park will be made freely available for use by members of the public.
An Aldi spokesperson said: “We have started investigation works on the former Yorkshire Bank site in Chapel Allerton.
“In the New Year we will begin the demolition of the existing building with the aim to start construction in the summer.”
The site was previously owned by supermarket chain Morrisons after they secured planning permission in 2013. But no work took place and after the site was put up for sale last year, Aldi snapped it up.
Initial plans were put forward earlier this year but planners said they were likely to be rejected unless the store was built towards the front of the site, facing onto Harrogate Road, with car parking at the back.
However after a site visit and presentation by the applicants, members of the Plans Panel agreed that positioning the store at the back would be acceptable.
The plans include a 76-space car park and access off Harrogate Road, as Leeds City Council is creating a new pedestrian crossing across the busy thoroughfare.