The long-running presenters, 50 and 63, have been left frustrated after executives backed Frediani, despite a months-long inquiry into complaints over his management style.
The probe, which began in June, examined claims made by staff and counter-complaints directed at the main hosts.
While Frediani will remain in charge of BBC Breakfast and the News at One, both Munchetty and Stayt are said to be "furious" at the outcome and insiders claim a departure would not come as a surprise, according to a new report.
A source told The Sun: "He makes no pretence of the fact that he is old-school and that means some people won't like his working practices, but they are delivering excellent results and that's what matters most to him.
"He feels like it has been a witch-hunt and it's taken its toll but he's cracking on. Naga and Charlie are furious and feel like the BBC have backed him over them.
"No one would be surprised if they ended up leaving."
The Standard has contacted BBC for comment.
In June, BBC News reported that Frediani, who has been in charge of the morning show since 2019, would take an extended period of leave after claims made about his behaviour.
The news service also said a HR adviser from consultancy firm PwC was supporting the corporation while it looked into the culture of the morning TV show.
It followed reports from The Sun and Deadline on an internal investigation into allegations of bullying.
New reports say a full investigation is not expected to be launched into Frediani, who accepted a Bafta in May when BBC Breakfast: The Post Office Special scooped the news coverage prize at the TV awards.
Two sources have said Frediani has been cleared of bullying allegations, according to US publication Deadline, with another source claiming that work examining the conduct of presenter Naga Munchetty continues.
This comes after The Sun claimed she had been reprimanded by bosses over two alleged incidents, which include an allegation of bullying, in three years.
A BBC spokesperson said: "While we do not comment on individual HR matters, we take all complaints about conduct at work extremely seriously."
In April, BBC executives Tim Davie and Samir Shah vowed that "today is the day we draw a line in the sand" after an independent review found some "well-known names" are "not being held to account for poor behaviour".
The workplace culture review, led by management consultant Grahame Russell from Change Associates, was launched in the wake of the furore over disgraced former newsreader Huw Edwards.