Figures from an unofficial survey carried out by the BBC show that Evans brought in an additional 650,000 listeners in February, The Mirror claims.
It is thought that both Evans and the corporation will avoid speaking publicly about the programme's performance until official Rajar figures are released in May. However, they are expected to show that the newcomer has an average daily audience of 8.6 million.
A BBC source revealed: "Everyone is delighted. Bosses were geared up for an initial slump - as often happens when any new presenter takes over - followed by a surge.
"No-one, not even Chris, imagined he would bring in so many new listeners so quickly. That he has drawn level, and now overtaken, Terry's figures is staggering. He is doing a great job in attracting younger listeners without alienating Terry's loyal followers."
Meanwhile, a spokesperson for the corporation said: "We are very much looking forward to receiving the Rajar figures when they come out in May."
Evans made his breakfast show debut at the beginning of the year after Wogan decided to stand down from the slot.