Dudek, 34, joins Pepe, Christoph Metzelder and Javier Saviola as new arrivals at the Bernabeu and will fill the gap left by Diego Lopez, Madrid's back-up goalkeeper who departed for Villarreal this summer.
However, that did not scare Dudek off from making the move and he revealed his delight at being handed the chance to join one of the world's biggest clubs.
"Before I played for big clubs like Liverpool and to be honest when I left Liverpool I thought I would never play for as big a club again, and never with such good players as I did at Liverpool," said Dudek.
"But when I finished my contract and heard that Real Madrid were interested in signing me first of all I was surprised but at the end I was thinking 'yes, I am going to do it', because Real Madrid might need an experienced goalkeeper like me and also need someone to be competitive with Iker Casillas.
"To be honest it was a very easy decision to make because all of my Spanish friends at Liverpool all said you cannot say no to Real Madrid."
Dudek revealed that he sought the advice of his former goalkeeping coach at Liverpool, Jose Ochotorena, after finding out about Real's interest.
"When I first heard that Real Madrid were interested in signing me I straight away called goalkeeping coach Jose Ochotorena, who used to play for Real Madrid," the former Feyenoord custodian said.
"He knows everything about the club and he said to me: 'Jerzy, don't wait, don't think about it. It is a fantastic club and if you get the chance to go there, it will be no problem for you. I know you, you can do it.' His advice was very simple."
