He expressed his excitement about joining the Merseyside club.
The extra 30 minutes failed to produce more goals, pushing the game to penalties.
Barcelona were close but failed to score four goals and level the game on aggregate, as Atletico won 4-3.
The 18-year-old Spaniard edged out fierce competition to be crowned the best young player in world football for the second year running.
Madrid remains the world’s most valuable football team, now worth $6.75 billion.
Girona put in a solid display to hold fellow Catalan club and La Liga leaders Barcelona to a 0-0 draw, the second consecutive time the Blanquivermells have taken a point in competitive matches at Camp
The semi-final represents a chance for Real Madrid to get revenge for their 3-1 loss two weeks ago in the Spanish Super Cup against Barcelona.
Bale made his professional debut with Southampton in 2006, joined Tottenham a year later, where he shot to stardom. He joined Real Madrid in 2013 in a world record £85.1 million transfer.
Modric joined Los Blancos from Tottenham in August 2012 and has made over 400 appearances for the club.
Real Madrid received one of the greatest shocks in Spanish football history 12 months ago when losing to the same third-tier opponents at Estadio El Collao.
