"In the meantime, we urge all customers to keep their personal information safe and if they are suspicious of any messages they receive to please give us a call to confirm the legitimacy of the message."
He wanted to remind everyone that legitimate messages from any bank would never ask customers to supply personal information, login details or second-factor identification.
If a customer received a text message from ASB, they should not click on the link and simply delete the message, he said.
The bank had updated its social media channels and website advising customers about the smishing campaign.
The ASB Contact Centre is available on 0800 803 804 if a customer is concerned about any suspicious messages they receive.