The 29-year-old who can operate as a winger or striker, has completed a medical and signed a four-year deal with the Whites, running until the summer of 2024.
Rodrigo spent time at Celta Vigo and Real Madrid during his youth career, before being snapped up by Portuguese giants Benfica.
He is no stranger to the Premier League, having spent time on loan with then top-flight side Bolton Wanderers during the 2010/11 season, making 17 appearances, helping the Trotters finish in 14th spot.
On returning to Benfica he scored 27 goals in 67 league outings and in the 2013/14 season helped the club win the treble, by securing the Primeira Liga, Taça de Portugal and Taça da Liga.
Rodrigo also helped Benfica reach the UEFA Europa League final in 2013 and 2014, playing in both finals against Chelsea and Sevilla.
Ahead of the 2014/15 season, Rodrigo joined Valencia for the campaign on loan, before his stay at the Mestella was made permanent.
In his six seasons with Valencia, Rodrigo made a total of 220 appearances in all competitions, scoring 59 goals.
He helped the side lift the Copa Del Rey during the 2018/19 season, scoring what proved to be the winning goal in the final as FC Barcelona were defeated 2-1.
Last season he played for Valencia in the UEFA Champions League and scored against Chelsea in their 1-0 victory at Stamford Bridge in September.
At international level, Rodrigo helped Spain lift the Under-21s European Championship and has won 22 caps for the senior side since 2014, scoring eight goals.
He could add further to that tally over the forthcoming international break, after being selected in the Spain squad for their fixtures against Germany and Ukraine.