A crowd of hundreds of protesters backing the government, carrying signs in English that said "genocide will not happen" and "stop interfering in Burundian affairs", greeted the diplomats as they landed in the capital Bujumbura, an AFP reporter with the council said.
At a meeting planned for tomorrow, the council hopes to persuade President Pierre Nkurunziza to agree to an African Union proposal of 5,000 peacekeepers, which his government has branded an "invasion force".
AU Commission chief Nkosazana Dlamini-Zuma said in a statement that there was an "urgent and imperative need for a strong unity of purpose" to solve the crisis.
Discussion of the peacekeeper deployment is expected to be a key element of talks at the AU summit in Ethiopia on January 30-31.
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An AU deadline for Burundi to accept the force has long passed with no action yet taken to deploy the peacekeepers, named the African Prevention and Protection Mission in Burundi (MAPROBU).
But the protestors who had been waiting for the diplomats made clear their opposition.
"Rwanda, stop Burundian refugees military recruitment," another protestor's sign read.
Relations between Rwanda and neighbouring Burundi are tense, with Bujumbura accusing Kigali of backing armed rebels and political opponents of Nkurunziza. Rwanda has denied all the claims.
Burundi descended into bloodshed in April when Nkurunziza announced his intention to run for a controversial third term, which he went on to win in July elections.
Hours before the UN diplomats arrived, Burundian rebels named a fugitive ex-general who fled after leading a failed coup bid last May as their leader.
Ex-general Godefroid Niyombare, a former intelligence chief, is the leader of the Forebu rebels, said spokesman Edward Nibigira, himself a former senior police general.