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In an email to customers, the online retailer said it was "working closely with Visa on a potential solution".
The move had been due to take effect on 19 January with Amazon blaming "high fees Visa charges for processing credit card transactions" when it was announced last November.
Amazon's latest message to customers did not say that the dispute had been resolved for good.
But it said: "Should we make any changes related to Visa credit cards, we will give you advance notice."
Visa also released a brief statement, saying: "Amazon customers can continue to use Visa cards on Amazon.co.uk after 19 January while we work closely together to reach an agreement."
It had said at the time the online giant announced the ban that it was "very disappointed that Amazon was threatening to restrict consumer choice in the future".
The online retailer was said to have been particularly angry at a rise in so-called interchange fees - additional cross-border costs - which it believes have risen five-fold since Brexit.
Such fees, payable by Amazon itself or merchants on its platform, would squeeze margins - or raise prices if passed on to customers.
Losing the right to be used for payments on the retail platform in the UK would have been a major blow for Visa.
Research by consultants Mintel published last year showed 89% of British consumers shopped on Amazon.
Susannah Streeter, senior investment and markets analyst at Hargreaves Lansdown said of the dispute: "The two sides have not completely backed down but last ditch talks over the weekend appear to have been productive and certainly Amazon is coming across as a lot more conciliatory in tone.
"Higher fees being charged by Visa remain a bugbear, and it's likely that a long term solution will involve some movement here, but it's not in either companies' interest for a war of attrition to re-start, with the prospect of significant losses in UK business for either side.
"This is a niggling headache Visa will want to see lifted as it grapples with competition from start-ups and more established rivals."