El Mundo Deportivo says Barca are lining up Enrique as a long-term successor for veteran Eric Abidal.
Enrique, the subject of intense interest from Liverpool, has a year left on his Toon contract and will be sold if he continues to refuse new terms tabled.
The defender did his chances of moving to the Nou Camp no harm when declaring: "Without doubt, Lionel Messi. Between Cristiano Ronaldo and Messi, I would sign Messi, of course. He has all the conditions to be the best footballer in history."