Snowflake has been the subject of other high profile breaches in recent yearspublished at 11:42 BST
Joe Tidy
Cyber correspondent
The company has been linked to incidents targeting services including Ticketmaster and Santander.
It is, however, unusual that any potential data breach would be revealed quite so publicly - and for customers to be informed in this manner.
Most extortions and negotiations by cyber criminals are conducted in private, with hackers hoping their discretion will result in a quiet pay-off.
Dan Bird, from cyber security firm Horizon3, says the pop up message apparently sent by the criminals implies their access has gone beyond the Snowflake database.
"Sending a push notification to Asos's app users would require access to the company's notification system, which is separate from the Snowflake data platform the attackers claim to have compromised," he says.
"If both claims hold up, it suggests the attackers got hold of credentials that opened more than one door."
