"On Sunday or Monday I will fly to Spain to pass medicals, on the same day I will sign a contract and will be presented as a new Real player. I hope so and I am really happy with this," Dudek told Polish newspaper Rzeczpospolita.
The 34-year-old guardian is a free agent after being released this summer by Liverpool, where he spent seven seasons.
Dudek claims that he received a host of offers recently, but when the Real bid showed up he felt that it was too good to turn down.
"The offer from Real overshadowed all the others," said Dudek.
"I was considering a return to Feyenoord, Jacek Krzynowek was encouraging me to join him at Wolfsburg, an interesting offer from a respected newly promoted Premiership club appeared and I was also thinking of moving to Portugal."
The Polish international is unfazed by the presence of Spain's number one custodian Iker Casillas at the Bernabeu, saying that he vows to fight for his place.
"I know that Casillas plays there. He is a great goalkeeper and, moreover, he is a club trainee. Fans love and respect such players. However, you can always find a place for yourself,"
"No player who has ambitions will tell you that he will be satisfied with sitting on the bench. I have gone through something like this at Liverpool and I know what kind of feeling it is. Why do you already give me such a role?"
Dudek struggled to find a place between the Reds' sticks in the past couple of seasons under Rafael Benitez, who usually preferred compatriot Jose Reina.