Trump has declared that “everything is under control” when all experts have said that the damage may take months or years to calculate.
People familiar with the matter have said the hackers were believed to be working for the Russian government.
The hack is so serious it led to a National Security Council meeting at the White House on Saturday.
The attempted intrusions, many of which were internally flagged by Microsoft Corp over the summer, were carried out by a group often nicknamed "Fancy Bear."
US officials have filed a glut of indictments against hackers in Russia, China, and Iran, levying sanctions, and issuing several warnings about state-backed digital intrusions.
The Kremlin did not immediately respond to a request for comment on the hacking of Fox's personal email.
The sources, who spoke on condition of anonymity because a law enforcement investigation is underway, said the hackers accessed the account multiple times between July 12 and Oct. 21 last year.
Secretary of State Mike Pompeo also called out the same GRU unit in February for conducting a cyberattack against the country of Georgia.
California-based Area 1 Security identified the hacking of Burisma and linked it to Russia's Main Directorate of Military Intelligence, or GRU.
Naama Issachar’s lawyers said their appeal was based on “an expectation based on the principles of law and justice that this decision will not stand.”