Justice secretary Jack Straw, communities secretary Hazel Blears and Olympics minister Tessa Jowell said the presenters should pay the fine out of their own pockets.
Straw said: "It is wrong that licence-fee payers will have to pick up the bill for this. It is ridiculous that the penalty will be paid by the public."
Jowell reiterated Straw's point. She said: "I think it would be honourable for Jonathan Ross to offer to pay it himself."
Blears, who was speaking on the BBC's 'Any Questions' radio programme, said: "The BBC is funded by all of us as licence-payers, so why are we having to pay the fine? Maybe Jonathan Ross and Russell Brand should pay it ... that might be quite a good idea."
The money will come out of the BBC's general budget and will be paid to the Treasury.
Ofcom said it did not have the powers to fine individuals. The media regulator released a statement last Friday saying that the "failures" by the BBC had led to breaches of two rules of its Broadcasting Code.
The first, of £80,000, was imposed because there was deemed to be an unwarranted infringement of privacy.
The second, of £70,000, was as a result of a failure to apply "generally accepted standards" to the content of the radio show "so as to provide adequate protection for members of the public".
The fines apply to two editions of Radio 2's The Russell Brand Show, the original on October 18 and another on October 25.
During the broadcasts, Brand and Ross left obscene messages on actor Andrew Sach's answerphone about Brand sleeping with his granddaughter, aspiring actress and burlesque dancer Georgina Baillie.
On one of the messages Ross shouted "he f***ed your granddaughter". Ofcom described this incident as "particularly gratuitous" and said it was compounded by the "prolonged nature" of the presenters' discussions about Baillie.
The incident, which caused a public outcry last October, became known as Sachsgate and resulted in the resignation of Brand, Radio 2 controller Lesley Douglas and the station's head of specialist music and compliance David Barber. The BBC also punished Ross with a three-month unpaid suspension.
Public complaints about the incident, with the Daily Mail leading a backlash by some national papers, rose to more than 44,000 -- 1,939 to Ofcom and 42,851 to the BBC.
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