Stoilov resigned soon after a 1-0 home defeat by Montenegro on Tuesday, four days after Bulgaria were thumped 4-0 by England at Wembley.
"I've already informed the Bulgarian Football Union's management and the team of my decision," Sirakov told Bulgarian state agency (BTA).
Bulgaria captain Stilian Petrov is also considering his international future, according to Stoilov.
Petrov replaced all-time leading scorer Dimitar Berbatov as captain in May, a week after the Manchester United striker announced he was leaving the national team.
"Stilian also said he might quit," Stoilov said soon after leaving his post. "But I hope he'll change his mind and I also want Berbatov to return. Bulgaria need them."
Petrov quit international football in 2006 after a row with former coach Hristo Stoichkov, only to return five months later. He has since refused to speak with local media after receiving some strong criticism.
Bulgaria visit Wales on October 8 in their next qualifier and the BFU has already began their search for a new coach.
"The problem is that we don't have much time as we meet Wales next month," said BFU president Borislav Mihaylov. "It's too early to say who'll replace Stoilov, we may appoint someone on a temporary basis."
Mihaylov added: "It's not a good idea to appoint a foreign coach."
"We can't afford a famous foreign coach. But you can't be sure with it. Look what happened with Russia, they have a very expensive coach but still they're losing at home too..."
