Vuk Draskovic made the comments in an interview with Serbian newspaper Blic.
He said his country's position was "more than grave" after the EU suspended talks because of the failure to deliver Gen Mladic to a UN tribunal.
The former Bosnian Serb military chief is charged with genocide over the 1995 Srebrenica massacre and other crimes.
He is thought to be hiding somewhere in Serbia. The European Union wants Gen Mladic to be transferred to the international war crimes tribunal in The Hague to stand trial.
'Replace security chiefs'
"Mladic must be arrested, and not any time, but in the next 10 days," Mr Draskovic said. "After that, it will be too late."
He told the daily that nationalists within the security service who were loyal to former President Slobodan Milosevic were responsible for the failure to find Gen Mladic. He said they should be sacked.
Meanwhile, President Boris Tadic said he hoped Gen Mladic would be arrested by 10 May, the Associated Press news agency reported. This would be the day before the cancelled EU talks had been due to be held.
The EU suspended talks on closer ties with Serbia on Wednesday after a weekend deadline for the arrest of Gen Mladic came and went.
On Friday, Serbian police arrested two people suspected of helping Gen Mladic evade capture, bringing to 10 the number of people arrested in recent weeks in connection with the search.
Serbian Prime Minister Vojislav Kostunica said a day earlier that the detentions had broken Mr Mladic's network of support and left Gen Mladic "all alone". BBC