The BBC and its licence fee payers will have to "bear the costs" of the inquiries in the wake of the Jimmy Savile scandal, "however much they are", Lord Patten has told MPs.
The BBC Trust chairman said the Pollard Review, which is examining a shelved Newsnight report into Savile's abuse, is expected to be completed by Christmas, but a second review led by Dame Janet Smith, looking at the culture and practices of the BBC during the years in which Savile worked there, could take much longer.
Appearing before the Commons Culture, Media and Sport Committee as part of its examination into the BBC's response to the Savile disclosures, Lord Patten said it was impossible to predict how much the reviews, which were ordered in the wake of allegations over Savile, would cost.
He said they were expecting the Pollard report next month, "certainly before Christmas", but said the second review would take longer, because of a "huge email trawl" that needs going through.
Asked about the cost of the inquiries - which he said would be met by the licence payer - Lord Patten said: "We don't yet know, but they will clearly be expensive, partly because of the number of lawyers involved.
"The Pollard inquiry has a QC who does the questioning at each session and I am told that QCs don't come cheap. I don't see how we could conceivably set down a capped cost without seeming to cap the work of the inquiries."
He added: "Other institutions which allowed Jimmy Savile to work and operate have not yet agreed inquiries into what actually happened, even though it's a couple of months since the allegations were made.
"We think we have gone about this in exactly the right way and I am afraid we must bear the costs, however much they are."
Lord Patten said the full Pollard Review would be published.
Acting director-general Tim Davie said the money for the report was coming from a contingency fund. "One of the things we feel very strongly about is that it should not impact on programme-making areas," he said.