The two neighbours have traded accusations over Rwanda's alleged backing of Burundian opposition members supposedly poised to overthrow the government in Bujumbura.
Burundi has also expelled several Rwandan traders from its territory, heightening the diplomatic tension.
However, following the death from heart attack of Nkurunziza last week at the age of 55, Rwandan President Paul Kagame ordered that all national flags in Rwanda and the East African Community (EAC) be flown at half-mast in solidarity with neighboring Burundi.
A statement to this effect issued by the prime minister's office quoted a directive from President Kagame as saying that Rwanda was “in solidarity with the government and people of our neighboring sister Republic of Burundi during this time of grief due to the demise of His Excellency President Pierre Nkurunziza".
The national flag and that of the East African community on Rwandan territory has been flying at half-mast since Saturday, 13 June and will be maintained until Nkurunziza's burial.
His successor who was elected to replace him has been cleared by the country's National Assembly to take over the reins of power.
But even in his death Kigali and Bujumbura continue to be at dagger's drawn, swapping allegations about supporting each others dissidents.
Both countries share the same culture and tradition.