The Saints, fourth in the Premier League after beating Arsenal 2-0 on Thursday, host second tier Ipswich Town on Sunday.
"I remember when I was a young kid watching the FA Cup final from Holland and I always wanted to play some sort of part in the competition in my career," Koeman told reporters on Friday.
"Cup final sounds like a special day. I remember speaking to Ruud Gullit about his experiences with Chelsea and it sounds like something I'd love to be involved in.
"It's a long way to Wembley and you have to beat a few teams to get there, but it all starts on Sunday."
Southampton made a superb start to the season, confounding their critics who predicted a difficult campaign following the departures of several leading players.
They lost five games in a row before Christmas but have halted the slump and are now back in contention for a top-four finish and a place in next season's Champions League.
Koeman has bolstered his squad with the signing of Dutch international Eljero Elia on loan from Germany's Werder Bremen.
The 27-year-old has been compared with controversial Liverpool forward Mario Balotelli having been involved in several arguments with fans on social media.
"If I had doubts about his private life I wouldn't sign him," Koeman told the BBC.
"It's the press and rumours, it's not true. We need really good players who want to develop themselves and be professional here.
"He is happy because he was looking for the next step of his career because he has not been playing in Germany," Koeman added.
"He is a good option for us and it can be very good for both parties.
"I believe in his qualities. He is fast, can play from the left and the right, and can take people on, one against one. He is hungry and that is the most important thing."
Southampton have also agreed a contract extension with Japanese defender Maya Yoshida to keep him at the club until 2018.
(Reporting by Ed Osmond; Editing by Toby Davis)
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