The 27-year-old made his Australia debut under Michael Cheika in 2019 but a catalogue of injuries limited him to four further tests until last July, when Schmidt handed him the captaincy for the first of three tests against Wales.
Injury again prevented him from playing any further part in that international season, however, and he managed just two Super Rugby games for the Reds this year, his eighth season at Ballymore.
"Following constructive discussions over recent months, Wright and the Reds were unable to finalise terms for a new agreement and will part ways amicably," the Reds said in a statement on Monday.
Wright played 86 games for the Reds, more than half as captain.
"I've had good days and bad days here but I've loved every second of it," Wright said.
"I'm ready to chase the next phase of my career ... I guess it's important for me to find new ways to challenge myself."