Regulator Ofcom imposed the fine for what it called the "gratuitously offensive, humiliating and demeaning" prank calls, broadcast last October on Brand's Radio 2 show.
"I was quite surprised when I read about this, that Ofcom were fining the BBC. The BBC is funded by all of us as licence-payers, so actually, are we having to pay the fine?" the Communities Secretary told BBC Radio 4's Any Questions.
"Then I thought maybe Jonathan Ross and Russell Brand should pay it... that might be quite a good idea," she went on - winning loud applause from the studio audience.
She spoke out after calls, some from opposition politicians, for Ross to pay the fine from his own wage - estimated at £6 million a year.
"This money should come out of Jonathan Ross's salary so that broadcasting does not suffer as a consequence of this error," Liberal Democrat spokesman Don Foster said.
Ofcom said the fine, a record for the BBC for a single case, reflected the "extraordinary" nature and seriousness of the BBC's failures and breaches of the Broadcasting Code.