This was announced by the company on Thursday, a possible sign of interest in the bank ahead of a planned relisting.
Abil said in a statement that a firm called Deep Value Investments had made unsolicited offers to some preference shareholders, without giving further details.
African Bank has 13.5 million preference shares listed in Johannesburg, according to Reuters data. Trade in all of its securities was halted after the central bank rescued Abil in August, putting it under outside supervision.
The unsecured lender had been hammered as its core market of low-income borrowers failed to repay their debts.
[DATA ABL:Abil] said the offer by Deep Value had nothing to do with its own restructuring plan, which is expected to see the bank relisted in Johannesburg early next year.
Reuters was unable to contact Deep Value Investments.
In a bankruptcy, holders of preference stock receive payment ahead of ordinary shareholders. Preference shares usually pay a fixed dividend and don't allow owners to have voting rights in a company.
Abil's preference shares last traded at 7.8 rand in early August, down nearly 90 per cent on the year.