"We need to improve upon what we've achieved already and keep progressing as a team," said the Spain international.
"Of course, we've got to do well in the Champions League and, well, we should aim to win it. Why not? I don't think that is out of the question."
Silva said their mentality had partly been forged by the FA Cup final victory last season, which ended a major trophy drought spanning four decades.
"It was a key season for us. We won our first trophy for 35 years and qualified for the Champions League, claiming a place among the best teams in Europe," he told the official Champions League magazine.
"We're up there with the elite now and this is enabling us to attract winners. These players know how to win, must win and accept nothing else.
"Winners don't believe in fate like others do. They simply cannot accept defeat.
"The UEFA Champions League is certainly where we want to be and the level we have been aiming for."
The 25-year-old believes he has now adapted to English football's top flight.
"It's now getting a bit easier for me," added Silva, who left Valencia in the summer of 2010.
"When you experience playing in two different leagues, you learn to adapt; you pick up elements from the best virtues of each culture.
"You adapt to the speed of the Premier League and you combine that with your Spanish instincts, to spend time on the ball, what you've learned about reading the game.
"Football in England is much more direct than in Spain - not just in terms of tactics or style, but in the overall approach.
"Even when you're winning 2-0, the crowd are urging the team to get the ball up to score a third."