Jasmina Stojanov, the president's press aide, said: "The police took one of the bombs from the man, but he is still holding another in his hand".
She said the man gained access to the building via an entrance normally used by journalists and visitors, at around 1000GMT.
A police official said that the man was a "psychiatric case".
"He sent his message to 11,000 email addresses this morning, threatening to blow up himself unless the government agreed to an out-of-court settlement of his claims," he told the Reuters news agency.
The presidency declined to comment on the incident, and it was unclear whether Tadic was present in the building.