The Spanish midfielder joined Bayern Munich from Real Madrid last week, making his debut in Saturday's 1-1 Bundesliga draw with Schalke.
Alonso won last season's Champions League with Real, although he was suspended for the final victory over local rivals Atletico, and also lifted the trophy with Liverpool in 2005.
The 32-year-old says he intends to continue his impressive run in Europe's elite club competion, and could equal the haul of Seedorf, who won the tournament with Ajax, Real and Milan.
"I've already won the Champions League with two teams," he said at a press conference on Monday. "It'd be a dream come true to win it with a third."
Alonso revealed his switch to the Allianz Arena came as something of a surprise.
"The whole transfer happened very quickly," he added. "It's an honour an a privilege to be here in Munich.
"I was surprised that I went straight into the team but I felt very good out on the pitch and my team-mates helped me a lot.
"I've followed the Bundesliga for the last few years so it's fantastic to be here now. It's a new experience for me.
"Munich is a great city. There's always lots to do here and I'm looking forward to discovering it with my family."
