The French goalkeeper spent last season on loan at West Ham, where he made 18 appearances in four different competitions.
Areola agreed to sign a five-year contract with West Ham after they triggered an £8m option to sign him on a permanent basis.
"I feel good, I feel great, I'm happy to sign and happy to stay here permanently and I can't wait to start to train and to do the job," said Areola.
"The main thing is that I felt the love of the supporters last season. Something for me that is really important is to feel that I'm loved. I know that I have to do my job and to do everything to make them happy, and obviously also the team, my teammates, the staff and Xavi Valero as well. The vibe of last season was great, so I just wanted to do everything to stay here.
"I feel comfortable, I feel like when I'm signing in a club, I want to feel that it's like my second family. Obviously, we're here every day and we are sharing lots of times together, so I have to feel comfortable with everyone and feel them comfortable with me as well, so I just feel comfortable here."