The 29-year-old, who has been at Camp Nou his entire senior career, has 18 months remaining on his contract.
And, with Manchester City boss Pep Guardiola said to be keen on reuniting with the Argentina international at the Etihad Stadium, Barcelona fans have become impatient with the impasse over a new deal.
"We have to remain calm," Enrique said, as quoted by AFP.
When asked about recent rumours linking Messi with a move away from Barca, he added: "We're not getting involved in that.
"The rhythm and the way this is worked out will be dictated by the club and Leo Messi himself.
"I don't know about numbers, I know about football. We all want to enjoy watching Messi here for many more years, nothing else matters for me."
Messi has been in fine fettle this season, netting 13 goals and providing three assists in 14 appearances in the Primera Division.
His figures are even more impressive in the Champions League, having scored 10 times in just five appearances.