This follows the decision by Church Procurator, Paul Reid KC, to drop several bullying charges against her, despite finding sufficient evidence for a "realistic prospect" of conviction under church law.
While Bishop Anne welcomed the outcome and expressed her intent to resume her duties, four bishops, including the church's Primus, Most Rev Mark Strange, have voiced concerns.
In an open letter, they questioned whether she remains the right leader for the diocese.
Bishop Anne condemned the intervention, calling it "ill-considered and inflammatory" and accused her colleagues of acting in an "unprofessional and un-Christian manner".
She revealed that three of the bishops had privately issued an ultimatum for her to step down, giving her only 24 hours to respond, despite her being on holiday.
In a firm response, Bishop Anne told the BBC: "I will not allow them to pressure me into quitting a role I cherish, nor abandon a Diocese that has shown me overwhelming love and support."
She also claimed to be the target of a "campaign of bullying" by a small group within the church.
According to the news outlet, Graham Robertson, diocesan chancellor for the SEC in Aberdeen and Orkney, criticised the bishops' actions as "ultra vires," or beyond their legal powers, and argued that their letter violated employment law and human rights legislation.
Bishop Anne, who had been suspended since August 2022 while the allegations were investigated, is expected to return to her role on 24 October.
A separate investigation by the Scottish Charity Regulator into diocesan accounts is ongoing.