Barclays, Halifax, HSBC and Lloyds customers should remain vigilant after fraud messages were issued.
Online customers are being contacted by 'banks' about savings accounts as the tax-year draws to a close.
The messages ask account holders to verify suspicious transactions.
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One fake message appears initially as a security pop-up asking for confirmation of a payment made from a digital device not used before.
Others ask the recipient to tap a link to confirm payment to a named person.
However, the messages contain fraudulent links that request sensitive information such as online banking details and full names, putting the person at risk of theft and banking fraud, reports the Mirror.
Speaking on bank fraud scams, Katherine Hart, a Lead Officer at CTSI, said: "I am witnessing so many reports of this scam; indeed, I have received multiple versions of it on my phone.
"The public is very vulnerable to this type of fraud, especially when more people rely on online payments.
"Fraudsters change the form and methods of their scams to match shifting consumer behaviour.
"The surge in online shopping and payments means that the public must be more vigilant when making online payments and receiving messages claiming to be from their bank.
"If you receive a suspicious text like this, please contact your bank directly and verify with them. Also, forward any scam texts to 7726, which is a free reporting service ran by Ofcom. We must protect ourselves and others from these scams but also provide vital intelligence to authorities."
To report scams, contact Action Fraud. To report email scams, contact the National Cyber Security Centre (NCSC) by emailing report@phishing.gov.uk.
