Ancelotti guided Napoli to a second-placed Serie A finish in his first season in charge last time out, with Insigne scoring 10 goals in 28 league appearances.
But Napoli is fourth on the table after seven matches in 2019-2020, already six points behind leader Juventus, while one goal in Insigne's past five league games represents a disappointing run of form.
The Italy international admits his relationship with Ancelotti is unusual but has denied suggestions the coach has played him out of position.
"My rapport with Ancelotti is a bit odd, as at times we have had rows, but that's all down to our respective personalities," Insigne said.
"There is a difference of character between us. We happen to squabble, but Carlo remains a great coach and I hope to win with him.
"He has not imposed a role on me I did not want since he took over at Napoli.
"Sometimes, when I get over-tired or anxious at training, it can happen that I respond to his observations, but these are things that end there. I often joke with the coach."
Insigne also sent a message to any Napoli fans doubting his commitment to his hometown club.
The 28-year-old, four times a Serie A runner-up with the Partenopei, said: "Some Napoli fans haven't understood me, as they consider me arrogant, as if they have a different image of how I am in reality.
"I hope that they can change their minds and stand by me, because I'd get myself killed for the Napoli jersey.
"The time has come for Napoli to win. We’ve been waiting too long now."