The Treasury-backed bank National Savings & Investments has been accused of a series of errors dating back years, which may result in a payout amounting to hundreds of millions of pounds.
In the region of 37,000 bereaved families claim they did not receive money that was rightfully theirs when a family member died, according to reports.
Complaints against the bank state that NS&I withheld Premium Bond prizes from the families of deceased savers, with some reporting that the bank delayed payments and lost track of investments.
Others claim they were forced to use lawyers, incurring additional legal costs, in order to recover the money they were owed.
Pensions minister Torsten Bell is due to address the issue in a statement to MPs in the House of Commons today.
The pensions minister is likely to face questions as to whether taxpayer funds will be used to help NS&I compensate customers for alleged failures in the management of their cash.
Treasury officials are understood to be working with NS&I to determine the exact amount owed to customers.
NS&I, which offers a range of savings and investments to more than 24 million customers, including over 22 million Premium Bonds holders, is yet to comment on the size of its exposure to redress and compensation.
Zoe Gillespie, investment manager at RBC Brewin Dolphin, said the scale of the problem remains unclear.
"The NS&I is currently working through a £3bn modernisation programme which is years behind, so there appear to be some issues with potential tech or customer service problems," Gillespie told the BBC's Today programme on Thursday morning.
NS&I need to "get on the front foot" to restore investor and savers' confidence, she added.
The bank’s digital transformation project, designed to modernise NS&I's operations and measurably reduce its running costs, has been heavily criticised by both the National Audit Office and the Public Accounts Committee.
The NAO said in November that the programme lacked a “realistic integrated plan”, while PAC warned in February that the host of challenges and delays to the project had exposed the taxpayer to additional risk.
NS&I issued an apology on Wednesday to anyone suffering a bereavement who had not received the customer service they should expect.
A spokesperson for NS&I said: "We recognise that dealing with bereavement can be challenging and would like to apologise to anyone who has not received the customer service from NS&I that they should expect, particularly at such a sensitive time."