The Sun reported that it was offered the images, which it said included shots of the duchess's children, Prince George and Princess Charlotte.
The paper said someone had contacted it asking for Â£50,000 within 48 hours.
The Met Police said inquiries were ongoing and no arrests had been made.
The haul from the hack is believed to include 3,000 pictures, and an anonymous person has reportedly tried to sell them via encrypted messaging service WhatsApp.
The Sun quoted a spokesman for Miss Middleton, who was maid of honour at her sister Catherine's 2011 wedding, as saying: "Thank you very much for drawing this to the family's attention.
"I can confirm that not only have the lawyers been informed but the police are about to be involved as well."
A Met Police spokesman said: "Police have received a report concerning the alleged hacking of a personal iCloud account. Specialist officers are now investigating."
In the summer, Miss Middleton and hedge fund manager James Matthews confirmed their engagement, with a wedding planned for next year.
Several high-profile figures have had images stolen from their iCloud accounts, including actress Jennifer Lawrence and singer Rihanna.
In July, US man Edward Majerczyk pleaded guilty to running a phishing campaign to steal private pictures and videos from film and TV stars, in what was known as the "celebgate" affair.
In 2014, Apple said it had expanded its use of "two-step verification" checks to protect data stored on its iCloud servers.
The process works by introducing an extra step after an account holder has typed their username and password into a device they have not used before.
They are also required to enter a four-digit code that is either texted to a trusted mobile phone number or sent via Apple's Find My iPhone app.
If the person does not enter the code, they are refused access to iCloud and are blocked from making an iTunes, iBooks, or App Store purchase.
Police are investigating claims an iCloud account reportedly belonging to the Duchess of Cambridge's sister Pippa Middleton has been hacked and private photographs stolen.