The 21-year-old becomes Sam Allardyce's seventh signing of the summer.
Enrique, who will wear the number three shirt at St James' Park, said: "I believe very much in the manager at Newcastle United and I believe they are going to be one of the top teams."
The Spanish club announced on Saturday they had reached an agreement with the Magpies for the transfer of the Spain Under-21 international.
And Enrique told his new club's official website that Allardyce helped make his mind up.
He explained: "I met Sam for dinner and I was very happy to listen to his plans for Newcastle United.
"He treated me very well and he understands me. He knows I have to get used to a lot of new things, like changes in culture and language, but he says he is going to help me adapt to that.
"He says it will make me stronger and he will have patience with me. I am really happy to hear that.
"I know Newcastle already have players with a lot of character and players who are used to winning many things and playing in Europe like Michael Owen, Mark Viduka, Alan Smith and Obafemi Martins.
"It will also help me to have people like Nolberto Solano and Albert Luque to speak to. Albert says he is going to take care of me at Newcastle."
Enrique joins Mark Viduka, Joey Barton, David Rozehnal, Geremi, Alan Smith and Claudio Cacapa as Allardyce recruits this summer.