The email reportedly mocked the cancer-suffering mother who also acts as her sons' agent.
Manchester City have responded to the allegations by releasing a statement promising a full investigation
"Following allegations made to the football club by Dr Anthonia Onuoha, subsequently reported in today's [Monday] media, the club can confirm that a board-led review has been launched into the matter," said the statement.
Dr Onuoha told the Sun: "When I opened my emails and saw the message, it was the worst day of my life, even worse than being diagnosed with cancer.
"I couldn't understand how anybody could behave like that. I just cried and cried for hours. I'm critically ill and at that point I was undergoing chemotherapy.
"I was just so shocked but I couldn't tell Nedum or any of my family because I didn't know how they would react."
Dr Onuoha said she replied to Cook the following day, writing: "Thanks very much for your insightful email."
She told the Sun: "The illness I have is very stressful, I'm having chemotherapy and radiotherapy, including having radiotherapy on my brain and this year has been traumatic. I didn't want anybody adding to my stress but that is what they've done.
"The feeling of humiliation still persists. Now I just want the FA and the Premier League to see what's happened and to take the appropriate action. Treating people like this is completely abominable."
Cook apologised for "any hurt or embarrassment this may have caused" as well as denying sending the email.
He told Dr Onuoha: "I cannot and will not condone this unacceptable behaviour. I can confirm Brian did not receive this email and disciplinary action is currently under way regarding one of our employees."