The bank wrote to applicants of the Bounce Back Loan Scheme (BBLS) in their personal name rather than business name.
Banks administering the initiative have been told by the government not to demand personal guarantees for the cash.
Bounce back loans are part of a package of measures to help struggling companies cope with the impact of the coronavirus.
Small businesses can receive up to £50,000 through the scheme, with more than one million firms accessing the funds.
HSBC alone has approved almost 140,000 of the loans, worth around £4.55bn.
The lender said it will contact people who received the loan in their own name, and they will not be personally liable.
A HSBC UK spokesman said: “We are aware of an administrative error that has impacted a small number of our bounce back loan facility offer letters.
“We apologise for the confusion this may have caused and would like to reassure any impacted customers that they will receive a restatement letter shortly.”