Although Royal Bank of Scotland group, the bank's parent company, said the initial technical glitch had been fixed, customers have reported a range of online services being unavailable.
The problems began last week when a software update for NatWest's payment processing system reportedly corrupted, also affecting services at RBS and Ulster Bank.
The problems saw bank balances not being updated or completely inaccessible, leaving people unable to complete purchases or withdraw cash.
NatWest has 7.5million personal banking customers but it is not clear how many have been affected by the technical problems.
Yesterday more than 1,200 branches opened on Sunday for the first time, with the same number opening between 8am and 7pm today.
Susan Allen, director of customer services at RBS group, said bank balances should largely be 'back to normal' today.
'The knock-on effects of this technical failure mean there will be bumps in the road,' she said.
'We will do everything we can to minimise further disruption to our customers.'
Customers voiced their frustration on the NatWest website this morning.
Chris Latimer from Liverpool wrote: 'wow first time since the glitch, i cant access online banking. this is not good.'
Jean from Brighton added: 'why can I not log into online banking to check my balance?'
Others said they had 'lost faith' in the bank and revealed wages had failed to arrive, while users also took to Twitter and other social networking websites to complain.
In a message to customers and staff this morning, Stephen Hester, chief executive of RBS, 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.'
He continued: '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.'
Mr Hester added: 'I also want to reassure customers that no one will be left permanently out of pocket as a result of this, and again, they should contact us directly about this.'
It has been claimed that some NatWest customers could have their credit score hit by the technical glitch.
The bank promised it was working with credit reference agencies to prevent any disruption after the software problem left many customers without wages and unable to pay bills or access their accounts.
However, affected customers were warned it is their responsibility to check for anomalies on their accounts and credit reports.
Kevin Mountford, of MoneySupermarket.com, said: 'It is understandable that customers are angry, especially if they are unsure whether salaries have been paid into their accounts, or important bills have been paid.
'NatWest has assured them that any fees incurred as a result of non-payment of bills will be refunded but they need to keep an eye on their statements.
'If you have incurred costs elsewhere, such as credit card charges, then you will need to provide evidence of this.'
NatWest customers are advised to speak to explain the issue in advance to mortgage, rent or credit card providers if payments are due.
Independent financial adviser Yvonne Goodwin added: 'If NatWest sort this out quickly then consumers should not need to worry about their credit status. But it's good housekeeping to check.'