Long queues were seen outside some of the Co-op’s convenience stores from Sunday amid the snow, with some shoppers asked to use cash.
Co-op and Morrisons said customers were no longer experiencing problems but they, and ACI, were studying the cause.
One MP said the problem exposed the risks of letting cash use “wither”.
ACI, which provides real-time payments processing for the retailers, said: “We are working closely with the IT teams at our partners to resolve the problem as quickly as possible. We apologise to shoppers for any inconvenience caused.”
The issue comes as contactless payments have taken off in the UK during the pandemic, with fewer consumers using cash to pay for groceries.
Customers complained about the issue on social media.
A Co-op spokesman told the BBC: “All card transactions are being processed as usual and our payment process partner is investigating after we experienced an intermittent issue.
“We would like to apologise to customers for any inconvenience caused during that time.”
The BBC witnessed the card processing issue affecting some of The Co-op’s stores meant that self-service checkouts had to be closed, requiring customers to queue to be served at tills manned by staff.
At some stores, customers queuing outside were warned on Monday evening that transactions had to be “cash-only” due to the ongoing issue.
Some customers said they had to use the convenience store’s cash machine to withdraw money to pay for purchases.
However in other stores, the problem was intermittent, impacting some payment card brands, but not others.
Shadow economic secretary to the Treasury Pat McFadden said: “This shows the dangers of letting the cash network just wither away as use declines.
“The government promised legislation to secure nationwide access to cash a year ago. It hasn’t been brought forward.”