She will be the first permanent female host of the flagship programme if she takes the role.
"We are in advanced negotiations with Zoe to replace Chris," a BBC source told the paper.
"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."
The 47-year-old has experience hosting the show, having stood in several times for Evans. She has also hosted a Saturday show on the channel since March 2017.
Evans caught listeners off-guard when he announced in August that he was leaving the morning show for a slot on Virgin Radio.
He was one of the BBC's most popular hosts and attracted an average weekly audience of 9.3 million after taking over as host from Sir Terry Wogan in 2010.
The 52-year-old has rejected suggestions that his decision to leave the BBC was motivated by money, although the BBC's director general has said that the broadcaster lost top talent after it published their salaries.
A BBC salary report in 2017 revealed that the popular presenter was the BBC's highest earner, taking home up to £2.2m.
The same report said that Ms Ball earned between £250,000-£299,999.
There has been speculation that the broadcaster wanted to hand the iconic breakfast show to a female presenter, following a furore about gender pay inequality at the BBC earlier this year.
A BBC spokesperson said: “We are considering and talking to a number of people. We haven’t made a decision and we’ll make an announcement when we have.”
Press Association contributed to this report