According to The Sun, the TV host was suspended last week after a "row".
But the BBC did not confirm the story, merely saying he was "not currently in production" and that it "wouldn't comment on individual staff matters".
The Sun quoted Wonnacott as saying he was "not doing Bargain Hunt at the moment for personal reasons".
His last Twitter post was on Friday, when he uploaded a photo of himself outside the BBC's central London headquarters.
"Outside Broadcasting House in London today for a dawn (well not quite !) meeting," he told his followers.
The antiques expert's phone went unanswered when the BBC News website attempted to contact him on Wednesday, and the nature of his alleged "bust-up" with Bargain Hunt's production team is unclear.
Wonnacott, a former director of Sotheby's, has been the dapper host of the popular BBC One programme since 2003.
He was a contestant on last year's Strictly Come Dancing series and recently presented The Great Antiques Map of Britain on BBC Two.
Guest presenters have been recruited to shoot the remaining half of Bargain Hunt's current series, the BBC said.
The future of Bargain Hunt presenter Tim Wonnacott is unclear after a newspaper reported he had had "a bust-up" with the BBC show's producers.