The 21-year-old has left United and signed for Harrogate Town on a free-transfer, with United negotiating a sell-on clause should he move for a fee in the future.
Harrogate's assistant manager is former United midfielder Paul Thirlwell, whilst former Blades youngsters Terry Kennedy and Lloyd Kerry are also at the CNG Stadium.
Wright, who didn't make a senior appearance for United but was a regular goalscorer for the U23 side, wrote on Twitter: "I would like to say a massive thank you to all at Sheffield United for the past six years, it's been truly unforgettable.
"The support shown from the staff, players and fans has been ace and I can't thank you all enough! Special mention to Korn and medical team that have given me rehab to get me back on the pitch as soon as possible!
"I'd also like to thank the club for not holding me back and giving me a chance for a fresh start. Once a Blade, always a Blade."