The Liga giants have been eliminated in the last four of Europe's premier club competition for the last three campaigns, at the hands of Barcelona, Bayern Munich and Borussia Dortmund.
Real hold the record for European Cup and Champions League wins having lifted the iconic trophy nine times in their history.
However, they have not tasted success in the tournament since 2002 - a run Alonso, who won the Champions League with former club Liverpool in 2005, is eager to end in this season's final in Lisbon on May 24.
When asked about regrets in his illustrious career in an interview with Club del Sportista, the 32-year-old - who has also won the FIFA World Cup and two UEFA European Championships with Spain - insisted that he does not have any "truly bad memories" in the game.
But, along with losing the Liga title on the final day of the 2002-03 with first club Real Sociedad, Alonso admitted that Real's failure to reach the Champions League final again ranks as his deepest disappointment.
"In recent years, playing for Real Madrid, being knocked out of the semi-finals of the Champions League was very painful," he said.
"(That) is why our aim is to reach the final this year and put those defeats behind us."
Real can take one step closer to their elusive 10th European crown by beating Schalke in the last 16 of this season's competition early in the new year.
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