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The company will alert users through email and push notifications when someone tries to change an account password, restore iCloud data to a new device, or when a device logs into an account for the first time, the report said.
Cook said Apple will broaden its use of two-step authentication to avoid future intrusions, the paper added.
Cameron Diaz has called the leaking of explicit photos and videos of dozens of female celebrities "a major violation".
Diaz, who stars in a new film about a couple whose homemade sex tape accidentally goes public, said, "I think that people really need to look at ... how would they feel if it happened to them?"
Speaking on ITV's This Morning, the actress said: "Whoever has done it, they will be caught and made examples of. This can happen to anyone. If these guys can do it to this group of people then everyone's vulnerable to it.
Her co-star Jason Segel called it "a terrible crime," adding, "It's very easy when you use the word 'celebrity' to take away the human aspect but there is an actual person with a life."
The leaking of explicit celebrity photos and videos was not the result of breaches in the security of Apple's computer systems, the company has said.
The US tech giant said none of the cases it had so far investigated were a result of its iCloud or Find my iPhone systems being hacked but as a result of a "very targeted attack" on the security procedures of celebrity accounts.
The leaking of the private photos of famous female stars sparked initial suggestions that Apple's iCloud service had been compromised to access the images and leak them across the internet.
The FBI and Apple are investigating the apparent hacking iCloud accounts that led to the publication of hundreds of alleged naked photos of female celebrities online.
Stars including actress Jennifer Lawrence have threatened to take legal action over the release of the photos.
Apple spokeswoman Nat Kerris said: "We take user privacy very seriously and are actively investigating this report."
The FBI said it was "aware of the allegations concerning computer intrusions and the unlawful release of material involving high profile individuals, and is addressing the matter".
The online hack which led to the posting of hundreds of explicit photographs of celebrities online could have been a password attack.
Hollywood stars including Jennifer Lawrence and model Kate Upton saw intimate photos posted on forum site 4chan on Sunday.
Initial reports concluded Apple's iCloud service had been compromised to access the images.
But a piece of computer code that repeatedly guesses passwords has been found online.
The script was posted to software site GitHub which apparently uses the top 500 most common passwords approved by Apple to try and gain access to user accounts.
If successful, it would give the hacker full access to the iCloud account, and therefore photos.
Apple have now apparently "patched" or fixed the bug. A posted update later said: "The end of the fun. Apple has just patched."
The technology giant is yet to make any comment on the incident.
Former Coronation Street star Michelle Keegan has said she is not a victim of a hacker who claims to have accessed hundreds of naked photographs of celebrities through their laptops and iPhones.
A list of 101 celebrity names, including Miss Keegan's, appeared on image sharing forum 4chan on Sunday with the poster threatening to publish intimate snaps online.
Hollywood actress Jennifer Lawrence has confirmed naked pictures posted of her are real.
A spokeswoman for Miss Keegan said she was aware of her client's name being on the list.
But she said: "There's no evidence that Michelle's phone or computer has been hacked and nothing has appeared online to suggest she has either."
Latest ITV News reports
Emma Watson has hit out at a popular "lack of empathy" towards the celebrity victims of a photo hacking scandal.
Celebrities including Hollywood actress Jennifer Lawrence have threatened to take legal action after naked snaps of them are hacked.