The senator, who was defeated on Sunday by President Luiz Inacio Lula da Silva in a runoff vote, said he intended to comply with the request, denying he planned to flee the country after the election.
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The younger Bolsonaro claimed, without providing evidence, that the police request was based on a ruling from Supreme Court Justice Alexandre de Moraes. Brazil's top court and federal police did not immediately reply to comment requests.
"I was ordered right on the street to hand over my passports, and I'm also prohibited from leaving the country," Bolsonaro said in a social media video.
He added Moraes' ruling was based on remarks from his younger brother Eduardo Bolsonaro, who said on a podcast last week that Flavio should leave the country if he lost the runoff election to Lula.
Lula won a historic fourth term in Sunday's election in a surprise result that defied polls. Bolsonaro refused to accept defeat and said he would ask for the result to be annulled.