A church that has asked to remain anonymous over safety fears, in the capital Kigali, has taken in six families so far according to Jenny (not her real name), a regular church-goer who is helping those seeking safety.
Violence erupted in Burundi last year when the president said he would remain in power for a third term.
Pierre Nkurunziza has been President of Burundi since 2005 and in 2015 he was nominated by his party for another term in office.
There has since be controversy from both his supports and opposition as to whether it was legal for him to run in a third term and protests erupted.
There was a coup attempt in May 2015, media was shut down and more than 150,000 people fled the country.
Since then thousands more have tried to find safety in other countries as violence intensifies and alleged killings continue of those thought to be supporting the opposition.
Jenny says the president's crackdown on criticism means people have nowhere to go: "So for as long as the president is there, he's still going to try and make sure that he stays in power.
"But the county's falling apart - people have lost their jobs, they're not working... so long as he's president I don't think there's going to be a change."
She told Premier thousands of refugees have crossed the border into Rwanda trying to escape the bloodshed.
She told Premier that not everyone in Rwanda has been welcoming their new neighbours.
"Generally the Christians are accepting, but human beings have a tendency to forget where they come from, and initially when the refugees had just come there were mixed feelings, they were like 'why are they here?' " she said.
"Then the community began to say, remember we were refugees once."
Rwanda suffered one of the worst recorded genocides of the twentieth century in spring 1994 when 800,000 people were killed in 100 days of bloody violence.
Jenny told Premier her prayer for all of those fleeing the chaos: "I hope that for the majority of them [refugees] they can just continue a life and continue school, even if they stay in Rwanda just that they can carry on not stop their education... so I really, really hope that they can find a home."
She told Premier many refugees are turning to Christ and are in desperate situations.
"My colleague who is himself a refugee was supporting a family of five with £2 weekly for food," she said.