Il Corriere dello Sport says Real Madrid see Allegri as a potential replacement for Alvaro Arbeloa, who is losing ground inside the board room.
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Defeat at home to Getafe on Monday night has Real now four points behind LaLiga leaders Barcelona.
Allegri is now being discussed amongst Real's directors, with his Milan contract only running to June, 2027. The Italian has long been a target for Real president Florentino Perez.
However, Allegri recently said: "There is total harmony with AC Milan. We are on the same page, working together. The club is planning for the future of AC Milan.
"I am happy to be here."
Despite his comments, there is a belief inside Real that Allegri can be tempted away this summer