The health and beauty retailer is asking permission to install a new shop front on Unit 3 in Meole Brace Retail Park.
The plans are for two new illuminated Superdrug signs.
It already has a shop in the town centre on Pride Hill.
Elsewhere on the retail park, plans to install a new shop front at the former Toys R Us unit is well underway.
Home Bargains, which already has two shops in Shrewsbury, is set to take over the store and create 50 new jobs.