The price cap prevents companies from immediately passing on wholesale costs to households, but the level is reviewed every six months.
It has already gone up by £139 on October 1, taking the average bill to £1,277, and will be reset again on April 1.
Cornwall Insight, the energy consultancy, has warned the level could rise from April 1 by as much as £400, while others have predicted higher increases.
Speaking on BBC Radio 4’s Today programme, Mr Brearley said: “It’s hard to tell by how much, but we do expect bills to rise in the future.
“We can’t predict everything and the wholesale market, as we have seen, has gone up and down extremely quickly.
“So we can’t predict fully what that will be, but looking at the costs that are in the system, we are expecting a significant rise in April.”
He added: “When we talk to customers, of course they are worried about rises in their bills. But the price cap is there to protect customers against unfair profits.
“But legitimate costs do have to be passed through. And unfortunately, we have seen an extraordinary rise in the gas price.”
Last night Kwasi Kwarteng, the business secretary, tweeted: “The Energy Price Cap is holding back a wave of instant bill increases. It will remain in place, and at the same level, this winter.”