TASS said the Supreme Court of the self-proclaimed Donetsk People's Republic (DPR), a territory recognised only by Russia and Syria, had received appeals from lawyers for Brahim Saadoun and Shaun Pinner.
It said another condemned Briton, Aiden Aslin, had not yet submitted an appeal, citing Aslin's lawyer.
All three men were sentenced to death last month for "mercenary activities" in fighting for Ukraine against Russia and Russian-backed forces, in what Western politicians decried as a show trial.
Their relatives say they were contracted to fight for the Ukrainian army, and therefore not mercenaries but regular soldiers entitled to the protection of the Geneva Conventions on treatment of prisoners of war.
TASS cited the DPR Supreme Court as saying the appeals would be considered within no more than two months.
It said Pinner had asked for his sentence to be commuted to life imprisonment.
An updated DPR criminal code published on an official website that took effect on Friday says the death penalty will start being used from 2025.
It is unclear what this means for the three men. The DPR, unlike Russia, has had capital punishment on its statute books since 2014, but had no legislation outlining how to enforce it until now.
On Thursday, the European Court of Human Rights (ECHR) said it had issued an order to Russia to ensure the two Britons did not face the death penalty.
Russia, whose parliament passed legislation last month to remove it from ECHR jurisdiction, said it was not bound by the order, and that the matter was in the hands of the DPR.
(Reporting by ReutersEditing by Mark Potter)
