She doesn't need a passport at all, only a valid ID card.
To return to the US, she only needs her Green Card
So yes, she can fly there and back without a passport.
SOURCE (since Tomas By spreads false info based on loose interpretations of non-authoritative info):
Timatic, the IATA database used by Airlines, says:
Bulgaria: Passport Exemptions: Nationals of Bulgaria with a national ID card.
US: Passport Exemptions: Passengers with a Permanent Resident/Resident Alien Card (Form I-551).
So again, the simple answer is she does not need any passport at all whatsoever.
She should Show her green Card and Bulgarian ID when checking in in for the USflight to Bulgaria, then the Bulgarian ID when entering/exiting Bulgaria. When checking in for the flight to the US, as well as entering the US, she should Show her green Card.