Griezmann goes into this season under a cloud after granting an interview in the summer in which he declared his desire to leave the club to play alongside "big players" at Atletico Madrid.
The deal subsequently collapsed, leaving Griezmann with relationships to rebuild, not least with the supporters.
He told the Diario Vasco: "I want to apologise again, turn the page and say that I regret what happened.
"On the pitch I will show my commitment to Real."
As for his reception from the Anoeta crowd, he added: "I prefer not to think about it. We'll see what happens.
"If they whistle again, I understand, but mostly I want them to encourage my team-mates."