The former Manchester United goalkeeper signed for the Viola on Friday.
Palladino said, "De Gea? His career speaks for him, he has international calibre.
"He also brings us a lot of experience in the dressing room, he made himself available by staying in Florence to work and from Monday we will start working together even if he has been out of action for over a year and therefore it will take time to get it back into shape."
On Sofyan Amrabat, after last season's loan with Manchester United, he added: "A great relationship was born with him because I didn't know him as a man.
"I met a great man and he gave me availability despite the transfer rumours so I threw him on the field (against Freiburg, 2-2) and he had a good performance. We'll see what happens but I'm happy to have him and I'd be happy to coach him all year."