Hackers are demanding a ransom from TalkTalk after attacking its website on Wednesday.
Chief executive Dido Harding received the email, but said she did not know whether or not it was genuine.
Ms Harding told the BBC: "It is hard for me to give you very much detail, but yes, we have been contacted by, I don't know whether it is an individual or a group, purporting to be the hacker.
"All I can say is that I had personally received a contact from someone purporting - as I say I don't know whether they are or are not - to be the hacker looking for money."
The phone and broadband provider said personal information like names, credit card details, dates of birth, email addresses and phone numbers were at risk.
TalkTalk said all customers could have been affected, but it was too early to say what data had been stolen. The Metropolitan Police said that no arrests had yet been made, and that the investigation was ongoing.
Ms Harding said: We brought down all our websites [on Wednesday] lunchtime and have spent the last 24 hours investigating with the Met Police.
"It's too early to know exactly what data has been attacked and what has been stolen.
"Potentially it could affect all of our customers, which is why we are contacting them all by email and we will also write to them as well."
It is the third cyber attack suffered by TalkTalk in the space on 12 months.
Ms Harding said: "Unfortunately cybercrime is the crime of our generation. Can our defences be stronger? Absolutely. Can every company's defences be stronger?
"I'm a customer myself of Talk Talk, I've been a victim of this attack."