Greenwood is considered one of the best young back-rowers in the Super League and has been on the radar of several NRL clubs.
Both Taia and Greenwood are awaiting visas before they officially join their new clubs.
"We wouldn't want to stand in the way of any player wishing to ply their trade elsewhere and Joe has been given a good opportunity at the Gold Coast,” Saints coach Kieron Cunningham said.
"It's always difficult to lose a home-grown player and personally I will be sad to see him leave.
“I've worked hard with Joe to develop his game and help him grow as a person - and he has done just that.
"Hopefully, one day after a successful NRL career, he will pull on the Red vee again."