Woodgate joined Real for £13.4m from Newcastle United in August 2004 but has made just one competitive start, against Athletic Bilbao last month, a match in which he scored an own goal before being sent off. He also scored an own goal in a friendly against Real Zaragoza last week.
But the England defender insisted: "I am just starting with Real Madrid, so why would I think about going? I feel a debt to this club because they have had confidence in me and still I have not been able to show my value to them.
"I have seen how things have got better and better for Beckham in Spain and how well things are going. I only hope that this can happen to me. I have not had much luck but I still have confidence in myself. I am looking forward to the challenge and mentally I feel strong. I am not going to get down about things that have been unfortunate."
Woodgate admitted that helping Madrid win the Champions' League would go some way to redressing the balance.
"[The club have won] nine European Cups and we all want the 10th, but now we have to think about getting three points against Rosenborg. We have a lot to play for," he added.
"I have played before in the Champions' League with Leeds and we faced Real Madrid. They beat us with a goal which was handled by Raul. He always reminds me about it - he says that it was legal but we always joke about it.
"For me it would be incredible and very special to play a final against an English team. There are very strong teams in the Premiership. Chelsea have a very solid side and are difficult to beat and Liverpool won the last Champions' League. That was very good for English football."
Real will have a bye from the third round of the King's Cup tonight, due to their participation in Europe, but a Madrid derby is still pick of the ties, as Rayo Vallecano and Getafe meet.
Getafe, who entered the top flight for the first time last year, have replaced Rayo as the third side in the capital behind Real Madrid and Atletico Madrid.
After achieving a respectable 13th-place finish in May they have undergone a transformation under former Germany international Bernd Schuster, who has led them to third in the table after seven games.
They lost their first game of the season against Real Betis 1-0 on Sunday but have impressed away from home winning three out of four games on their travels.
Rayo, on the other hand, have moved in the opposite direction, dropping out of the La Liga in 2003 and then down to the third division a year later. They are coached by former Spain and Real Madrid midfielder Michel.
The 23-times Cup winners Athletic Bilbao face a tricky trip to play Real Union in a game which could increase the pressure mount on the new manager Jose Luis Mendilibar.
The Basque side, who lost to eventual winners Real Betis in last year's semi-finals, are second from bottom in the league after suffering their fourth defeat in seven in their 1-0 home defeat to Seville on Sunday.
Atletico Madrid, who also fell at the semi-final stage last year, meet former La Liga stalwarts Las Palmas following their 3-0 defeat at home to Real Madrid at the weekend.Reuse content