After three campaigns spent acquiring experience in the competition - last season reaching the last 16 for the first time - Kompany believes it is time to come of age.
City will open their latest challenge against one of Europe's giants in Bayern Munich and the skipper reckons they belong in such company.
City have actually grown familiar with Bayern, having been drawn in the same group in 2011-12 and last term, but Kompany is relishing the latest tussle.
Speaking at a press conference to preview the Group E opener at the Allianz Arena, the inspirational defender said: "Every single time you come here is an opportunity to prove yourself.
"We now take it as a challenge against one of the top three clubs in the world
It is one of hardest places to come to.
"It's time to start making these games close and making up the gap between ourselves and the other top clubs."
With two Barclays Premier League titles in the past three seasons, City's development is continuing apace.
Establishing themselves at Europe's top table is now the next target and Kompany is well aware that expectations at the club are rising.
The 28-year-old said: "You start a season at Manchester City nowadays and it's not acceptable not to put all the trophies on your ambitions list.
"It's a case of me and all the players looking at all the trophies and thinking we have to go until the very end of the season.
"I mean, the club is asking us to win now, so that's one thing.
"We have come so far in the last four years, we have been successful, but you always want to take it a step further.
"Let's take it in context - we are not favourites by any means for this competition but we would like to compete and challenge."
With the experience of winning the Premier League now under their belts, Kompany even believes clearing the European hurdle could prove relatively more straightforward than initially breaking through domestically.
He said: "Before we won the league it was more difficult to believe we could do it than with the Champions League.
"My experience of the Champions League is that it's not more difficult to go to the end - but everything has to be perfect
The Premier League is just a ferocious battle to the end.
"Eventually it will happen our way."
City's progress in reaching the knockout stages last season was tempered as they were paired with Barcelona and lost out over two legs to the Catalan giants.
Yet they could have been facing lesser opponents had they managed to score one more goal in their final 3-2 win over Bayern to top their group on goal difference.
Kompany wants to go all out to take control of the group this season.
He said: "We know how much damage it can cause your hopes if you don't win the group
We always face difficult teams if we are not seeded but we were very close last year."
Manager Manuel Pellegrini will again have to give his team-talk ahead of the match in the hotel as he completes his two-match touchline ban for controversial remarks about a referee last term.
The Chilean, who turned 61 on the eve of the match, is forbidden from making any contact with players or coaching staff on matchday after they arrive at the stadium.
As he is also banned from giving a post-match press conference, he also opted not to speak to media before the game, spending his birthday overseeing a training session.
His assistant Ruben Cousillas, who will make decisions in the technical area, did speak, confirming that striker Stevan Jovetic has failed to overcome the hamstring injury that kept him out against Arsenal last weekend.
Defender Pablo Zabaleta is also missing through suspension after his sending-off in the loss at the Nou Camp last season, while midfielder Fernando still has a groin problem.
Source : PA