In most cases, the charities will not have been aware of the trading, because it is usually done on their behalf by fund managers.
But HM Revenue & Customs has deemed bond washing to be a form of tax evasion under legislation from the 1950s and is insisting that penalties be paid.
Bond washing involves selling a bond just before the interest or dividend is due, then buying it back within six months, once the price has fallen. This typically results in a profit for the purchaser and a capital gain for the original bond vendor.
HMRC officials agreed in a meeting with the sector last week that charities were not guilty of flouting the law intentionally. It has dropped a proposal that penalties should be backdated 20 years, which could have meant fines totalling millions of pounds.
The penalties will now be based on how much charities are deemed to have been involved in bond washing during the past six years.
John Hemming, head of tax at the health research charity the Wellcome Trust, attended last week's meeting. He said: "HMRC wants to agree a method for determining a charity's liability that will not involve too much of a burden. This will be based on an estimate that is determined in a way agreed by both parties."
More charities could be affected than was originally thought, because HMRC has asked about bond washing in common investment funds. One CIF has more than 48,000 investors.
The issue came to light last December, when HMRC asked charities for details of bond washing liabilities. This surprised sector tax experts, who considered the law was not relevant to modern trading systems, in which bonds are bought and sold electronically, without any direct relationship between the buyer and the seller.
HMRC said discussions were continuing. Helen Donoghue, director of the Charities Tax Group, said she hoped to issue a briefing note soon.