Chief executive Stephen Hester said "on this occasion we have let them down" and "this should not have happened".
Over 1,000 Natwest, RBS and Ulster Bank branches stayed open longer on Saturday and will also be open between 9am and noon on Sunday to deal with accounts affected by the technical hitches which left customers unable to pay bills.
Natwest said the computer error had been fixed and that it was working "round the clock" to clear the backlog of transactions.
The repercussions have extended beyond the banks' own clients, with one couple who are not NatWest customers left unable to move into their new home.
Hundreds of customers have vented their frustration over the problems, with some finding payments had gone awry and others observing disappearing wages, or holidays and home purchases disrupted.
Account balances were not updated properly overnight, meaning credit and debit payments were not showing up as quickly as they should.
In a statement, Mr Hester said: "I am very sorry for the difficulties people are experiencing.
"Our customers rely on us day in and day out to get things right, and on this occasion we have let them down. This should not have happened.
"Right now my top priority, and the priority of the entire RBS Group, is to fix these problems and put things right for our customers.
"This is taking time, but I want to reassure people that we are working around the clock to resolve these problems as quickly as we are able."
Meanwhile, concerns were raised over penalty charges, amid suggestions that thousands of customers could be hit with fines if their regular household bill payments - including mortgages - are affected.
The problems also extend to Royal Bank of Scotland (RBS) and Ulster Bank users.
All three banks are part of taxpayer-owned RBS Group, which has moved to quash fears over withdrawals, assuring customers their money is "in the system" and staff will be on hand in branches to help them access their cash.
It has pledged no one will be left out of pocket. Anyone who incurs penalties because of the problems has been urged to contact their branch.
In an email to account holders, NatWest said: "We apologise to all of our customers who have been affected by our current technical issues.
"These resulted in money credited to customers accounts overnight not appearing in their balances as it should.
"This problem is strictly of a technical nature, we have fixed the initial problem and our priority has been to work round the clock to sort the backlog as soon as possible.
"We appreciate this has caused an unacceptable level of inconvenience.
"Staff in our branches are ready and available to answer any customer questions and help with emergency needs."
The problem reportedly arose after staff tried to install a software update on RBS's payment processing system, but ended up corrupting it.