The Reds have allegedly tabled a £20m offer the talented playmaker, although the Saints are holding out for an extra £5m.
Lallana, who is part of England's World Cup squad, told reporters: "I'm ambitious. I have aspirations to play in the Champions League.
"Ever since I've played international football, you realise there's that other level to play at. It would be great.
"I went to a couple of the Champions League games last year and the feeling you get, the buzz, it's an unbelievable atmosphere. It's definitely a goal of mine to play in the Champions League."
The 26-year-old contributed nine goals and six assists for Southampton last season.