"Everybody should respect the borders of Burundi. If the troops are in violation of this decision, they will have attacked Burundi, and each Burundian must stand up to fight them. The country will have been attacked and we will fight them," Nkurunziza said in a speech broadcast on state radio.
The 54-member African Union gave Burundi a four-day deadline on December 17 to accept a 5,000-strong force to halt months of violence, pledging to send troops even though Burundi said it was opposed to an "invasion force".
"You cannot send troops to a country if the United Nations Security Council has not accepted it... The UN resolution says the international community should respect the independence of Burundi," Nkurunziza said in one of his strongest speeches yet following the unrest.
Burundi is still recovering from an ethnically-charged civil war between majority Hutus and minority Tutsis, which cost an estimated 300,000 lives between 1993 and 2006.
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The violence - which has seen hundreds killed and included an abortive coup, regular ambushes on security forces, street battles and even failed mortar bombings on the presidential palace - echoes attacks carried out during the civil war.
"But this is not the case here, because we are facing a security problem. It is not a political issue, because this was resolved by the elections."
Rebels last week said they had formed a force "to protect the population" and uphold the Arusha Agreement that paved the way to the end of the civil war but which they say Nkurunziza has violated by his controversial third term in power.
Earlier this month UN Secretary-General Ban Ki-moon said that deploying UN peacekeepers to Burundi was an option to quell the violence. AU Commission chief Nkosazana Dlamini-Zuma has written to the UN Security Council asking for "full UN support including the authorisation of a support package" for the force, the AU has said.