De Bruyne and Silva clashed after the final whistle as the former looked to raise a point to the fourth official - De Bruyne was stopped by Silva before demanding that the Spaniard let him talk.
City continued their unbeaten record in this season's Champions League group stages with a 2-1 win over their Italian opponents which took Pep Guardiola's side three points clear at the top of Group F.
"We just had a little discussion. A minute later, that's forgotten. I have those discussions with my wife too. You too, I suppose. Sometimes it's needed," De Bruyne told The Telegraph.
"Not everything is always positive. But this was something small. I wanted to ask something [to the fourth official] but I wasn't allowed to. But as I said, it's nothing to make a big deal out of. It happens sometimes."
Manchester City face Burnley at the Etihad on Saturday and are in EFL Cup action against Wolverhampton Wanderers on Tuesday.