City, the impressive leaders of the Barclays Premier League, can secure a place in the last 16 of Europe's elite competition with victory at Napoli on Tuesday.
That would represent another major step forward for the rapidly-advancing club - who ended a long trophy drought by claiming the FA Cup last season - but Silva does not intend to stop there.
The Spaniard told UEFA's 'Champions' magazine: "The Champions League is where we want to be and the level we are aiming for.
"Now that we have arrived, it is time to enjoy ourselves, make the most of it and show everyone what we can do.
"Last season was key for us. We won our first trophy for 35 years and qualified for the Champions League, claiming a place amongst the best teams in Europe."
He later added: "We need to improve upon what we have achieved already and keep progressing as a team.
"We must finish in the top four in the Premier League and secure Champions League football every season.
"It's not enough just to do it once. Of course, we've got to do well in the Champions League and we should aim to win it.
"Why not? I don't think that is out of the question."
City made a stuttering start in their European campaign by collecting just one point in their opening two matches.
They got back on track by completing back-to-back victories over Villarreal earlier this month to move second in Group A.
Despite that, Tuesday's encounter in Italy had long looked like being the key fixture and so it remains.
Victory over Napoli at the intimidating San Paolo Stadium will secure a place in the knockout stages but anything less could leave City needing to beat group leaders Bayern Munich in their last game.
Napoli are also still in the hunt for a top-two finish as they lie two points behind City heading into the penultimate round of matches.
Striker Edinson Cavani told reporters at the pre-match press conference: "We need three points if we want to keep going in this prestigious competition."
Cavani netted 26 goals in Serie A last season but has found it more difficult so far this term, scoring only six times in all competitions overall and just once in his last eight appearances.
However, the Uruguay international remains in optimistic mood, saying: "I'm feeling good, even if I haven't scored as many goals as I did last season.
"The team and the coach have faith in me and the amount of goals I score is not the most important thing, the most important thing is for us to win games.
"I like to take a positive attitude out onto the pitch and I work very hard in training, which enables me to play well and assist the team, even if I don't score."