The 21-year-old central defender has announced he wants to play in Europe, after impressing in the Copa America.
Villarreal, Real Madrid and Inter are also interested in Godin.
Godin revealed: “I trained with Nacional in the morning and then said farewell to my teammates and the everyday staff here.
“Something like this brings strange emotions and it all happened very fast. Overall I’m very happy because this is what I have always been waiting for.”