The Argentina international is contracted to the Blaugrana until the summer of 2018 but negotiations are yet to begin over an extension.
Recent reports have suggested that his father, Jorge, who works as his agent, will open talks with Barcelona following El Clasico on December 3.
It’s understood that former boss Pep Guardiola is keen to bring him to Manchester City, but Enrique has urged him to put pen-to-paper at the Nou Camp.
Enrique said: “We all know how much media [speculation] there is about Messi.
“We all know what he means to Catalonia, what he means to Barcelona. But I’m sure Messi, when he considers everything, will give the correct information.
“There’s really nothing I can say. Of course, I’d love him to finish his career here.”
Ajax are bracing themselves for a bid from Italian giants Roma for the signature of in-demand attacking midfielder Hakim Ziyech.
Defending champions can prevail again with goals on the cards in Kiev
Ex-Celtic man Murdo MacLeod says it is inevitable Kieran Tierney will leave the club eventually but is not expecting him to go soon.
Bournemouth have announced first-team pair Baily Cargill and Ryan Allsop will both be released when their contracts expire.