The Strictly Come Dancing: It Takes Two host has pipped fellow Radio 2 presenter Sara Cox to the job and is hammering out the details of a new contract, according to The Sun.
Evans recently took his millions of listeners by surprise when he announced he would be leaving the flagship programme and returning to Virgin Radio.
Ball and Cox were tipped as possible successors, while Jo Whiley, Simon Mayo and Dermot O'Leary were also among the runners and riders.
There has been speculation that BBC chiefs wanted to replace Evans, one of the corporation's highest-paid stars, with a female host following controversy over gender pay inequality.
The paper says Brighton-based Ball, who was favourite to take the reins at Britain's most-listened to show, is weighing up how to balance the five-day-a-week slot and family life.
She will be the first female full-time host of the show if she takes the role.
The 47-year-old mother-of-two has hosted a Saturday afternoon show on Radio 2 since March 2017 and has stood in for Evans on several occasions.
A BBC source told the paper: "We are in advanced negotiations with Zoe to replace Chris. A contract hasn't been signed and her new pay hasn't yet been agreed. But she was our number one choice and has been offered the job."
Evans announced he was leaving the coveted breakfast show for Virgin in August, raising the likelihood that he will take many of his listeners with him.
Since taking over from Sir Terry Wogan in 2010, he has attracted an average weekly audience of 9.3 million, rising at times as high as 9.9 million.
Evans, 52, has rejected suggestions that his decision to leave was due to money.
Earlier in the year, his BBC salary was shown to be between Â£1.6 million and Â£1,669,999 as the corporation revealed earnings of its top talent in its annual report.
Evans was the BBC's highest earner when the salaries of stars at the corporation were first revealed in 2017, taking home up to Â£2.2 million.
In the same report, Ball was in the Â£250,000-Â£299,999 bracket.