City have won the FA Cup and the Premier League since the 2008 Abu Dhabi-funded takeover that transformed their fortunes and making an impact on the continent is their next objective.
After exiting at the group stage of the Champions League in each of the past two seasons, City are now potentially within 90 minutes of reaching the last 16 for the first time.
Victory over CSKA Moscow in their latest Group D fixture at the Etihad Stadium would be enough to secure their route through if Bayern Munich also avoid an unlikely defeat to minnows Viktoria Plzen.
Pellegrini accepted qualification as a minimum requirement when he took charge of the club in the summer and recognises its importance.
He said: "For Manchester City it would be a really great achievement.
"We all know what happened the last two years - they couldn't get to the round of 16.
"For a club that is trying to grow every day, to go to the next round is very important."
Defender Pablo Zabaleta believes the time is right to make the leap, having witnessed all of the developments over the past five years at first hand.
Zabaleta joined the club the day before Sheikh Mansour took over and began the big spending that accelerated City's development.
The Argentinian said: "We know that the owners been spending much money to try and build a good team and try to have one of the best teams in Europe.
"Sometimes it takes time. We won the FA Cup and the Premier League, (but) the Champions League is a different competition. You play against the best teams in Europe.
"It is time for this club to take a step forward."
City beat CSKA 2-1 when the sides met on a poor surface in the Russian capital a fortnight ago.
@jimbobbarker (Torquay United) ...You do know how a league system works I assume? The Premier league is conducted over 38 games. Each team plays all the other sides in the division home and away. The rules deciding what happens should teams end level of points was in place at the start of the season. At no point were Manchester United ever 15 points clear, their greatest lead was eight points. For much of the season City were between five and seven points clear of United - using your logic, it could be argued "our slump" handed United the initiative, but that's something of an irrelevance, because the league is decided over the course of 38 games. Not 14. Or 23. Or 32. For the record, and as a long standing season ticket holder who took in the glory of League 1, or whatever it was called when reached those levels, the investment City received has in no way detracted from my enjoyment of our title success. Like it or not money has always made the football world go round - and if you don't believe that I suggest you review what was behind the success of clubs in the 1890's and early 20th century.- thecat