According to El Confidencial, Madrid president Florentino Perez offered an initial €6million with another €3m to follow for the highly-rated Valverde who plays for Penarol.
The report claims Arsenal did not match Madrid's price, only offering €3.5m for the 17-year-old who starred in the South American under-17 Championship.
He scored seven goals in nine games during the tournament in March.
GETTY
Federico Valverde (left) celebrates scoring for Uruguay during the South American under-17 tournamen
It adds that Penarol's president Juan Pedro Damiani was reluctant to see Valverde leave but the deal from Madrid was too good to turn down.
Valverde's signing is the latest young player Madrid moved for after the protracted transfer of 16-year-old Martin Odegaard in January.
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