The City captain and England defender was embroiled in an unsavoury altercation with Maupay during their Premier League clash on Monday.
Walker had to be calmed down by the referee and then his manager Pep Guardiola.
According to The Mail, an expert witness and forensic lip reader - Jeremy Freeman - has attributed the following quotes to Walker.
Walker is said to have approached the fourth official and shouted: "I'm gonna knock him (Maupay) out," to which fourth official Stuart Atwell replied: "Shush."
However, Walker went on to say: "Wait there," pointing to the tunnel, before walking back over to referee Jarred Gillet.
He then added: "Ref he spoke about my kids, he spoke about my kids, not once, but twice."
Maupay added: I didn't, it's OK."
Gilet turned to Maupay and finished: "Now, now, just calm down, listen to me... think about it, think about it."