The 52-year-old says he's happy at St Johnstone, amid speculation over his future.
Paul Williams left his role as caretaker manager last week as the Championship club begins its search to fill the permanent position, after Dougie Freedman was fired in the middle of March.
It was reported by The Daily Record on the weekend that Forest owner Fawaz Al-Hasawi wanted a British manager with a connection to the club, with Wright having spent four years at the Reds as a player.
Nevertheless, the 52-year-old claims he's happy with St Johnstone, reminding Forest that he has a long-term contract to fulfill.
“They are one of my former clubs and a club with a great history," he said via The Courier.
“But it’s just speculation. It’s nice to be linked with that but I signed a long-term contract here and I’m happy here."
Wright led the Saints to a Scottish League Cup final in 2014, eliminating Rangers in the semi-finals, while he also led Lisburn Distillery to the Northern Irish equivalent of that competition in 2011.
He has never managed in England before and, considering the names Forest had been linked with recently - like, say, Gus Poyet, who took the Real Betis job - the appointment of Wright might have been slightly underwhelming.