Stoilov, who has twice coached the Bulgarian national team, also managed Levski Sofia in 2006 when they became the first team from the Balkan country to compete in the Champions League group stages.
Kazakhstan have been without a coach since January when Russian Aleksandr Borodyuk resigned following a run of poor results. Kazakhstan lost six and drew one in 2017.
"My task is to raise the level of Kazakhstan," Stoilov, known as a tough disciplinarian, told reporters. "I'll give my best to help the team. The national team is the main team in the country."
Kazakhstan, which gained independence from the Soviet Union in 1991, have never qualified for the World Cup or the European championship finals.
They have previously employed several non-Soviet coaches, including Dutchman Arno Pijpers, German Bernd Storck and Czech Miroslav Beranek.
"I have been living in Kazakhstan for a long time and I can say that I know a lot about the country," Stoilov said. "It's clear that the Kazakhstan (team) have problems but we'll try to change the situation for the better."
Kazakhstan have been drawn alongside Georgia, Latvia and Andorra in League D Group 1 in the inaugural UEFA Nations League, the latest addition to the football calendar.
(Reporting by Angel Krasimirov; Editing by Toby Davis)
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