The 20-year-old has scored 27 times and registered seven assists in 56 appearances for his current club, while he netted on his debut for Uruguay against Bolivia on September 5.
Alvarez's contract with Penarol still has three years left to run, but it is believed that there is a €20m (£17m) release clause in the agreement.
According to AS, Real Madrid have been alerted to the youngster's progress and will continue to monitor his performances as they consider a move.
The report claims that Carlo Ancelotti's side could make an offer next summer if Alvarez enjoys another strong campaign for his club.
The striker, who would allegedly initially turn out for Real Madrid Castilla if he arrived in the Spanish capital, came through the youth system at Penarol before making his professional debut in September 2020.