A Bentley hit Vucic"s car in an inner Belgrade suburb on Saturday morning, Interior Minister Nebojsa Stefanovic told the state-run Tanjug news agency.
"Presidential security promptly reacted and those (three) people were held," he said.
"Every detail will be investigated and during a probe all facts relevant for the case will be established."
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For the time being there are no indications or evidence that Vucic was a target, the minister said.
The men will be questioned by prosecutors on Monday.
The three work as security guards in a Belgrade discotheque and have criminal records, the Politika daily reported.
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The paper said they borrowed the Bentley from a local football player who spent the evening in the discotheque.
Nearly a year ago Vucic, who was then prime minister, was moved to a safe location after weapons, including a rocket-launcher, were found near his house.
At the time the authorities labelled it "preparation for assassination," but a probe did not find evidence for that.
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