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Mr Vucic and Mr Blair met on June 19 and discussed Serbia's bid to become an EU state.
The meeting in a government office was opposite the ruins of an army base bombed in the Nato airstrikes.
Mr Vucic said: "We analysed the political situation, and I discussed our EU path."
When asked about sitting down with a former enemy, he said: "You can't live in the past all the time ... We are now part of the modern world - we have moved on."
He added: "I was impressed, even about his decency, because we were sitting in an office, and opposite us was that destroyed building in the centre of Belgrade, the office of the army general staff.
"He saw that, and he didn't say a single word because he thought it would be impolite to say something.
"I didn't want to say anything about it of course, it all happened 15 years ago."
Serbia is among a number of central and eastern European countries bidding to join the EU.
Mr Blair is also advising Albania on its EU candidacy, along with his former press secretary, Alastair Campbell.
Mr Blair is facing increasing scrutiny over his wide range of diplomatic, commercial and advisory roles.
Former Middle East ambassadors were among the signatories to of a recent letter calling for him to be sacked as the international envoy to the Middle East.
A spokesperson for Tony Blair said: "Tony Blair believes that the new Serbian prime minister is a strong reformer and he supports his programme.
"There is no project in, or contract with, Serbia."