Speaking to the Royal Television Society, the minister also suggested that "the BBC probably has reached the limits of reasonable expansion".
He said: "Although the Trust has performed better than its predecessor, I don't think it is a sustainable model in the long term.
"I know of no other area of public life where - as is the case with the Trust - the same body is both regulator and cheerleader."
He added: "There may indeed be a case for a smaller licence fee. But there is a proper timetable for determining that.
"One of the unbroken conventions adhered to by successive governments, to avoid the suggestion of political interference in or pressure on the BBC, has been to respect the multi-annual settlement system.
"I resolutely believe that to be right. Any attempt to break that convention would rightly be seen as a direct assault on the BBC's independence. "
Bradshaw also expressed his opposition to Trust chairman Sir Michael Lyons's plan to return £5.50 to each licence-payer in preference to 'top-slicing' the money to aid regional news from other broadcasters.
"Is this really about the long term interests of public service content? I would just like to point out that the £5.50 is not the BBC's to give away," he said.
"It was agreed on top of the current licence fee income for the BBC to fund help with digital switchover.
"However, Michael, if you want to return £5.50 from the BBC's share of the licence fee to the public - or more if you wish - let me know and I’m sure it can be arranged!"