Stoilov left his post after a disastrous start to their Euro 2012 qualifying campaign, which saw them defeated by England and Montenegro.
Aston Villa midfielder Petrov, meanwhile, previously quit international football in 2006 after a row with former manager Hristo Stoichkov only to go back on his decision just five months later.
But the back-to-back defeats have seen skipper Petrov consider his position once again, although Stoilov is urging him to carry on playing.
"Stiliyan also said he may quit," Stoilov said soon after his exit.
"But I hope he'll change his mind and I also want (Dimitar) Berbatov to change his mind. Bulgaria need them."