According to The Telegraph, the move comes because more taxpayers' money is to be injected into funding the banks' recovery.
Lloyds will receive a further £5.7 billion from the Government; and RBS will collect up to £25.5 billion more from the Government in a move that sees the taxpayers' stake in the bank climb from 70% to 82%.
But the news comes amid reports RBS is to shed a further 3,700 jobs.
The union, Unite, says ministers and employers have a duty to prioritise saving jobs over securing the best price for the banks' assets.
Unite national officer Rob MacGregor said: "Another day, another announcement bringing huge uncertainty to employees at the part-nationalised banks, RBS and the Lloyds Banking Group. We cannot allow a situation to arise where workers in bank branches in high streets and towns across the country are made to pay the price for the banking executives' recklessness.
"While this is a decision largely out of both banks' hands in divesting these assets, simply securing the best price would be letting down loyal and long-serving staff. Any potential buyers should be assessed on the commitment to job security and protection of terms and conditions, not short-term profit maximisation.
"The employers, UK Financial Investments and the Government all have a duty to these long-suffering staff to ensure that opportunistic buyers are not permitted to asset-strip these institutions, leaving thousands of them facing a bleak future. For Lloyds-owned Cheltenham & Gloucester staff this is a further blow and a return to massive uncertainty as the on/off decision on their future continues.
"The Government has saved the banks, now it is time to save bank workers."
