Terry signed a new five-year, £135,000-a-week deal with the club on Friday.
He said: “The Champions League is the unfulfilled dream for me.
“It’s the one I’d really like to win now. When I look at my trophies, the Champions League is the one that’s missing.
“I’ve won pretty much everything apart from that and I’ve also been lucky enough to realise virtually every ambition I’ve ever had.
“Two Premier League titles, the FA Cup, the Carling Cup, playing for England and now captaining them.
“So it’s one of my last remaining goals and the sooner the better.
“Football’s a short career and I don’t want to look back in 10 years and say I didn’t give it everything to win all I possibly could.
“I don’t want to be left there saying, ‘I wish this, I wish that’.”