Fewer than 20% of taxpayers currently being told they owe extra money to HM Revenue & Customs (HMRC) will be able to have the tax written off.
About 1.4 million people owe tax because of underpayments in financial years 2008-09 and 2009-10.
If an underpayment was due to a Revenue mistake, then it can be written off under a policy known as the extra-statutory concession.
But Revenue officials say only a minority involved mistakes.
The figure was revealed in evidence to the MPs on the public accounts committee.
"At the moment, only a minority of the underpayments are caused by Revenue error, which would make them eligible for writing off under the concession," said Sarah Walker, the Revenue's director of PAYE, self-assessment and national insurance.
"It is quite a small proportion - I think we would say it is less than 20% - and then of those, a proportion then will claim the concession and we will then write it off."
The Revenue explained that each request would have to be looked at on a case-by-case basis to see if the concession applied, but stressed that it would apply only in limited circumstances.
New computer system
The underpayments and overpayments were discovered when the HMRC introduced a new computer system to make the normal annual checks on people's tax liabilities and payments under the PAYE system.
This replaced the previous manual checks which had involved comparing information from 12 separate databases.
The new computer system highlighted the fact that in the 2008-10 period, more than six million people had paid the wrong amount of tax.
While some 1.4 million people are now receiving demands for extra tax totalling about £2bn, 4.3 million others will be receiving cheques to compensate them for previous overpayments amounting to £1.8bn.
A further 900,000 potential demands for tax arrears were scrapped by Treasury ministers because the sums involved were all for less than £300.
Ms Walker estimated that 70% of the letters being sent out in batches would be delivered by Christmas, with the rest being delivered by the end of January 2011.
The Revenue is hoping to collect £2bn from those who did not pay enough tax in previous years, though this amount may be reduced by successful claims from taxpayers that the Revenue was at fault.
Due to concern about the apparent unfairness of levying extra tax bills, the government has decided that those who have to pay more than £2,000 will not have to pay interest if they need extra time to pay.