The company, which has 16.5 million customers worldwide, said personal information, such as names and contact details, "may have been accessed" as a result.
However, it does not "believe that payment card information or account passwords, were impacted".
Customers received a mobile app notification on Tuesday, titled "Asos hacked", which directed them to a Telegram account.
The message read: "Dear ASOS DPO and IT, we have full compromised the Snowflake instance. Engage with us, or we will leak it," followed by the Telegram link.
On Tuesday afternoon, the company confirmed that an "unauthorised customer notification" had been sent out through its mobile app.
In a statement, the company said: "We are investigating unauthorised activity involving third-party platforms that we use to communicate with customers.
"We took immediate action to restrict access to the notification platforms and are working with our internal and external specialist advisers, as well as all relevant authorities.
"Our website and app are operating as normal, with no current disruption to any aspects of our operations.
"Customer trust is incredibly important to us, and if the situation changes an update will be provided as appropriate."
It is understood that the National Cyber Security Centre (NCSC), a part of GCHQ, has offered Asos assistance.
The notification message sent out by cyber attackers refers to cloud firm Snowflake, which stores data for many major companies.
Snowflake said it has "found no compromise" of its platform after launching an investigation following the notification message.
"The investigation is ongoing and we will provide further updates as soon as more information becomes available," a spokeswoman added.
Asos told shareholders it has cyber security insurance with a large provider and said it is "too early" to quantify any potential impact on its trading.
Shares in the company fell by more than 10 per cent on Tuesday as a result.
The UK is the group's largest market, representing 49 per cent of all revenues in the first half of the latest financial year.
It comes after raft of UK retailers were targeted by cyber attackers over the past two years, including Marks & Spencer and Harrods.
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