The Spain international will miss Saturday's top-of-the-table clash with City as well as Premier League clashes with Aston Villa and Everton after failing with his appeal against a Football Association charge for a stamp on Liverpool's Emre Can on Tuesday.
Costa was challenging for the ball by the touchline in the first half of the League Cup semi-final second-leg tie at Stamford Bridge when he forcefully stood on Can's shin.
The incident went unpunished during the match, which Chelsea won 1-0 to progress to the final 2-1 on aggregate, as referee Michael Oliver did not see the challenge.
Pellegrini urged the frontman to change his ways on the eve of the crunch clash at Stamford Bridge.
"He is a great player, a very important player," said Pellegrini. "He has his character, it is not an easy character but I hope for him, that this punishment, I am not trying to take advantage for the game tomorrow, I think it will be a good thing for him to change in the future.
"He doesn't need to do it in that way because he is a very good player, a top player."
City were criticised last week when they were eliminated from the FA Cup by Middlesbrough after spending the week in Abu Dhabi, but Pellegrini used his news conference on Friday to aim a dig at Chelsea, who were knocked out by League One side Bradford City.
"About last week, about Abu Dhabi, another club that didn't travel to Abu Dhabi also lost in the FA Cup," he added. "It was not the reason we lost.
"This week was a normal week and we worked well."
