The Mirror says with Slot under pressure, Liverpool chiefs have identified Enrique as the ideal replacement for the Dutchman.
And it's suggested Enrique would welcome an approach from the Reds should Slot fail to turn things around.
Enrique said on Cadena SER two years ago: "I follow above all the Premier because I would like to go to England to work.
"I wouldn't go to any team but to one that could do important things, which greatly reduces the number. I don't put my hopes up, really, because there are many candidates."
He added: "I am lucky because my personal life fills me up," he said. "That it takes longer, that no offers arrive, well, it is what it is. I would go to a team with possibilities."
Liverpool go to West Ham tomorrow having lost nine of their past 12 games.