Barclays has said the "technical issue" impacting payments and transactions for customers has been resolved, but is still working on updating bank balances for some customers.
The bank said delayed payments had been processed but that it was still "addressing any outstanding issues", following days of disruption.
On Friday when the outage began, customers told the BBC they were having trouble with mobile and online banking, and also making essential payments.
The technical issues were not caused a cyber attack, it is understood, but the bank has not explained what was behind the problem or how many customers were affected.
"We are very sorry for any disruption and will ensure that no impacted customer is left out of pocket," a Barclays spokesperson said on Sunday.
"We are keeping our call centres open for longer this weekend and we will be proactively contacting customers who may be vulnerable."
The bank said account balances would all be updated by the end of Sunday and that its services were back up and running as normal.
However, reports on the website downdetector, which monitors outages, were still in their hundreds early on Sunday, with most people experiencing problems with mobile and online banking.
Some customers on X also replied to Barclays saying they were still facing problems and asking for compensation.
Barclays is one of the UK's largest banks, with more than 20 million UK retail customers.
On Saturday, downdetector showed nearly 5,000 issues had been reported with Barclays, more than double the number reported on Friday.
On Saturday afternoon, Barclays' website said problems with its app and online banking were ongoing, and warned customers that they may face issues making and receiving payments.
The website also told customers that their high street branches may not be able to assist with all queries "due to issues we're facing".
Friday, which was pay day for many people in the UK, was the deadline for self-assessment tax returns.
Some Barclays' customers said they were unable to make their payments to HMRC.
A statement from HMRC said it was working closely with the bank to minimise the impact on those submitting self-assessments.
It confirmed that issues related to the Barclays outage would not result in late payment penalties as these do not apply until March 1.
