The national chain has submitted plans for 19 The Martlets in the town centre, which has been empty since HMV closed its doors.
Its planning application to Crawley Borough Council is for exterior signs outside the retail unit.
Dunelm sells bedding, curtains, furniture, beds and mattresses and has stores in Horsham, Shoreham, Rustington and Chichester.
To view the applications visit CR/2020/0186/ADV.
According to job listings posted on Indeed and GlassDoor the Crawley shop would be a brand new concept high street store called Dunelm Edit.
It says: “This will feature some of our bestselling product and will feature a new style look and feel as well as an enhanced digital shopping experience for customers.”
Dunelm has been approached for comment.
A message from the Editor, Gary Shipton:
In order for us to continue to provide high quality and trusted local news, I am asking you to please purchase a copy of our newspapers.
With the coronavirus lockdown having a major impact on many of our local valued advertisers - and consequently the advertising that we receive - we are more reliant than ever on you helping us to provide you with news and information by buying a copy of our newspapers.
Our journalists are highly trained and our content is independently regulated by IPSO to some of the most rigorous standards in the world. But being your eyes and ears comes at a price. So we need your support more than ever to buy our newspapers during this crisis.
Stay safe, and best wishes.