In a High Court conference room filled with hundreds of judges and lawyers who supported him, Mahmoud said he had told the president "I will not leave my home or my work unless it is by an assassination," al-Masry al-Youm reported.
A presidential spokesman said late Thursday Morsi had offered to appoint Mahmoud ambassador to the Vatican. Mahmoud said he had rejected the offer during a phone call Friday night, and that he considered the offer an attempt to remove him.
He said Morsi did not have the power under the constitution to remove him from office.
Protesters supporting Morsi's decision to remove Mahmoud demonstrated at the Supreme Court late Friday night. The protests degenerated into fighting.
The prosecutor has been a target of public criticism after senior figures who had been accused of organizing attacks on protesters while former Egyptian President Hosni Mubarak was still in power were acquitted, Ahram Online reported.
A number of judges had threatened to resign if Mahmoud were removed from office, but he vowed Saturday to complete his term.
Judges and prosecutors were often criticized during Mubarak's reign as being susceptible to government pressure, the BBC reported.
Notable deaths of 2014 [PHOTOS]
Ebola health workers killed in Guinea