The game was twice temporarily halted in the first half after England debutant Tyrone Mings alerted officials to the abuse directed at him and others from various parts of the Levski Stadium in Sofia.
As well as the monkey chanting, some Bulgarian fans made Nazi salutes during the game.
The president of Bulgaria's football association Borislav Mihaylov stepped down following the game after pressure from the country's Prime Minister Boyko Borissov.
The English FA faces sanctions too, over national anthem disruption and an insufficient number of traveling stewards.
World governing body FIFA has also asked UEFA to notify it of any punishment potentially handed down to the BFU.
"This would allow any sanctions imposed to be extended worldwide," said a FIFA statement.
CNN's Aleks Klosok contributed to the reporting.