Diamond resigned last week amid public anger after Barclays agreed to pay nearly 450 million pounds ($697.8 million) in fines for manipulating a key interest rate.
Newspapers have reported that Diamond could be in line for exit payments of as much as 17 million pounds.
"There isn't anything government can directly do about it, but I think in view of the shame that has already been heaped on Barclays bank, I would be very surprised if the chairman and the board were to allow another outrage to occur," Cable told BBC television.
"I would sincerely hope that the board of Barclays will take a fairly strict view about all of this," Cable added.
Barclays chairman Marcus Agius and another senior executive also quit last week after the bank admitted that its traders had rigged the London Interbank Offered Rate, or Libor, between 2005 and 2009. Libor, the rate that big banks say they borrow from each other, is used to settle interest rates on trillions of dollars of contracts globally.
The scandal has reignited anger in Britain against bankers, who are blamed for a recession the country is struggling to escape.
($1 = 0.6449 British pounds)
(Reporting by Tim Castle; Editing by Erica Billingham)
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