Núñez was born in Montevideo and started his playing career at the age of 19 playing for Nacional. His good form earned him a chance to play for Uruguay, and he was part of the squad for the 1959 Copa América.[2]
He moved to Spain to join Valencia, and he won the Fairs Cup in two consecutive seasons with the club. Later in his career he played for Mallorca and Levante.
Núñez worked as the manager of a considerable number of clubs in Spain and has also managed in Mexican and Uruguayan club football. He was manager of the Costa Rica national team in 1992 and the Uruguay national team between 1994 and 1997, where he led Uruguay to the Copa América 1995 championship.