The boss of TSB last night admitted the bank was on its knees after its disastrous IT upgrade as staff claimed branch computers were now seizing up.

Five days after 2million customers were left unable to bank online by the IT meltdown, workers claimed that internal software was failing too.

One employee said there was a growing backlog of uncleared cheques as customers resorted to other payment methods.

Yesterday 1million customers were still having problems with banking online as chief executive Paul Pester offered better savings rates in a bid to stop them leaving.

Paul Pester (pictured) offered better savings rates in a bid to stop them leaving the bank following the glitch

He faced calls to resign over the fiasco – and did not rule out standing down from Spanish-owned TSB, which has been dubbed 'Totally Shambolic Bank'.

Referring to the problems with new software, which had been run out of Spain, Mr Pester said: 'We think this is a fight worth fighting but we're down on our knees at the moment. I will do everything in my power to put things right for customers.'

Mark Brown of union TBU, which is not recognised by TSB, said problems with the computer programs in branches made it impossible to serve customers properly.

He said: 'The system isn't working in many cases, it's crashing all the time. When you get error messages they're in Spanish, so people have no idea what they say.

'The number of people coming into the branches is just not normal, and staff in the branches are having to deal with that at the same time as facing a system which is crashing and is slow.

'The idea the branches are working normally is just rubbish – people are coming in at 7am and going home at 8pm in tears because they can't do their jobs, and the more they can't do their jobs, the bigger the queues become.'

Mr Pester has persuaded the chairman of TSB's parent group Sabadell to give him control over the software issue and brought in experts from IBM to fix the problem.

TSB still does not know why some customers are unable to use their online banking.

Yesterday half of customers were still not able to access their web accounts, while one in ten could not get their funds using TSB's mobile banking.

Mr Pester insisted the system itself was working smoothly but had been overwhelmed by the numbers of customers logging in.

He said: 'The platform is running extremely well. This is down to a simple capacity problem.'

Mr Pester has persuaded the chairman of TSB's parent group Sabadell to give him control over the software issue

The chief executive said the bank will not charge any overdraft fees this month, costing the lender £10million. He also pledged to hike interest rates on the Classic Plus Account from 3 per cent to 5 per cent.

Mr Pester, who was paid £1.8million last year, refused to be drawn on whether he will give up his bonus this year and would not say if he was preparing to resign.

The Federation of Small Businesses (FSB) said it had faced questions from members unable to pay their staff because of TSB's problems.

FSB chief Mike Cherry said: 'Small business customers of TSB will have been significantly impacted by the unfortunate problems of recent days.

'Paying bills, transferring staff wages and checking that invoices have been settled are among the fairly basic functions of business banking which have been hindered.'

One branch worker told the Mail that the episode has been traumatic for TSB's 8,600-strong workforce.

Speaking on condition of anonymity, they said: 'It's horrendous.

'We have staff in the branch with a system that doesn't work, trying to help customers who can't use phone bank or internet and mobile banking – because that isn't working either.

'Branches have backlogs of customers' bills and cheques that haven't been processed yet and still aren't able to process – customers are expecting cheques paid in on Monday to clear by Friday but they haven't even been processed yet. God knows what's going to happen.'

Labour MP Wes Streeting, a member of the Commons' Treasury select committee, said: 'There are serious questions to answer.

'Cybersecurity is so high on the agenda that banks really do need to make sure they have good systems that are robust and workable.'