The German announced he is to leave Borussia Dortmund at the end of the season after a successful seven-season spell at Signal Iduna Park.
Coupled with the pressure surrounding Pellegrini amid City's stuttering form, speculation has mounted regarding Klopp taking the top job at the Etihad Stadium.
"It doesn't hurt - it happens in every club if you don't have the results you deserve," the Chilean, whose side sit fourth in the Premier League, said.
"When you manage an important club if you don't have the results the club deserves everybody talks about changing the manager.
"It has been the worst period for results, but last year there was more pressure.
"Nobody is happy with the results when you are not in the fight for the title.
"It's logical that the thing is to sack the manager but I don't think that is the correct diagnostic.
"We are thinking about the next game, the game we must play against West Ham on Sunday."
City, who host West Ham on Sunday, are 12 points behind leaders Chelsea, with fifth-placed Liverpool only four points adrift of the champions.
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