Pep Guardiola's side lead 2-1 on aggregate following a closely fought tie at the Etihad Stadium in which De Bruyne and Sergio Aguero scored second-half goals to cancel out Bobby Reid's first-half penalty for the Championship (second-tier) team.
De Bruyne's strike was his eighth of an outstanding season for the Belgium midfielder and Silva expects him to play another pivotal role in Bristol.
"Kevin, he was already a fantastic player, but he has definitely moved up to another level this season," Silva told the club's website (www.mancity.com).
"He is a fantastic footballer and has everything. He has done amazing things this season and I'm certain he has become one of the best players in the world, without a doubt.
"I've been fortunate to play with many great players, including Cristiano Ronaldo for Portugal. Kevin and Cristiano do not play in the same position, of course, but it is fantastic to play alongside him and David Silva because they are so good and so consistent."
Bernardo Silva joined City from Monaco for 43 million pounds ($59.8 million) last year but has been restricted to just 13 starts and four goals in all competitions so far.
The 23-year-old's opportunities have been limited by the excellent form of De Bruyne and David Silva in central midfield and Raheem Sterling and Leroy Sane on the wings.
"I'm feeling better and better, but I'm still not showing my very best form," Silva added. "I have to be patient and wait for my opportunities. It's my first year in England and I know next year I will do much, much better.
“I always try to improve and that won't change, but I think the City fans are yet to see the best of me, but will do in the months ahead and next season."
Gabriel Jesus, Fabian Delph, Phil Foden and Benjamin Mendy remain sidelined for Tuesday's trip to Ashton Gate, but Vincent Kompany could return from a calf injury.
(Reporting by Matt Westby; Editing by Christian Radnedge)
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