The company is to open in Cannon Street, Middlesex Street and Blomfield Street in September on the site of the former Flying Burrito outlets, purchased in July.
Bigg, who has worked on projects for Eat, Mitchells & Butler and Cantaloupe, will be responsible for determining the appearance of the stores, including signage, interiors and all artwork.
Wrap it Up! marketing director, George Groves, said: “We are delighted to be working with Julian on the new projects.
“The design and branding of our outlets is an important part of Wrap it Up’s ongoing development.
“The new look Wrap it Up! outlets will potentially be a template for the company as it continues its expansion.”
Bigg added: “Exciting things are happening at Wrap it Up! and I am pleased to be a part of it.
“I look forward to working with the company on this project.”