The Blues have enjoyed a memorable season, winning the domestic Double for the first time in their history.
However, they saw their Champions League hopes ended by former boss Jose Mourinho's Inter Milan and the European Cup remains the holy grail for Abramovich.
Terry insists Abramovich is driven to see Chelsea win Europe's premier competition and will not be happy until they do so.
"There is no doubt in Roman's mind and all our minds - we clearly want the Champions League," Terry told The Sun.
"He said we must get it next season even though he is delighted with the Double - and we're determined to do it.
"We've come so close over the last five or six years and, at times, have been the best team in the competition yet haven't gone on to win it.
"Next season, we want to move on from what we have done this time.
"We want to retain the Premier League title and win the FA Cup for the third year running.
"And, if we can add the Champions League to that list, the fans, players, management and Roman would be very happy."
Source: Team Talk
Source: Team Talk