The girls were reportedly ill and are in camps located in Chad, Niger and Cameroon, with one of them nursing a broken wrist.
The footage, not released publicly but seen by the London-based online newspaper The Mail on Sunday was taken in a jungle clearing a month after their abduction.
In the video, eight girls, dressed in their school uniforms of pale blue gingham, plead for release as they stand in front of a camera.
The newspaper said they were clearly scared, upset and trying to be brave as they walked in turn to a spot in front of a white sheet fixed to a frame between trees.
According to the publication, four of the girls could be heard in Hausa Language, stating that they were taken by force and that they were hungry.
One of them aged about 18 said tearfully, "My family will be so worried."
Another, speaking softly, said, "I never expected to suffer like this in my life.' A third said, 'They have taken us away by force" and the fourth complained that, "We are not getting enough food."
The video, allegedly taken by an intermediary on May 19, said the newspaper, was intended to serve as 'proof of life' for the girls and to Jonathan to accede to the terrorists' demands.
Two earlier videos showed the girls seated on the ground, dressed in hijabs, reciting the Koran.
In the videos, Boko Haram Leader, Abubakar Shekau, declared he would sell the girls into slavery or marry them off to their kidnappers if members of the sect in detention were not released.
Pressure from the international community and criticism of the Jonathan's slow response to the kidnapping had led to a series of contradictory pronouncements from his government.
Some ministers have declared that government would not negotiate with Boko Haram or consider the release of prisoners, while official spokesmen said "the window is always open for dialogue."
Last week, Chief of Defence Staff, Alex Badeh said the government knew the location of the girls and claimed that police and military had been 'following them' since the abduction. He refused to divulge details, saying it would put the girls in further danger.
The Mail claimed that Badeh's announcement may have been the result of government officials seeing the new, unpublished video and may have been able to persuade Boko Haram's intermediary to provide details of the location. It is believed the hostages have been split into at least four groups.
The report said one Dr Stephen Davis, an Australian who has advised three Nigerian presidents on how to negotiate with the country's militant groups, has spent the past month trying to help free the girls.
'The vast majority of the Chibok girls are not being held in Nigeria,' he said.
'They are in camps across the Nigerian border in Cameroon, Chad and Niger. I say the "vast majority" as I know a small group was confirmed to me to be in Nigeria last week when we sought to have them released.'
Saying the Federal Government has been engaged in negotiations with Boko Haram's spiritual leader Abubakar Shekau in a bid to secure the girls' release, the report quoted the Australian describing how fraught the negotiation process has been.
'One of that small group of girls is ill and we had hoped we might convince the commander of the group holding her that she should be released so we could give her medical treatment,' Dr Davis said.
'There are other girls who are not well and we have come close to having them released but their captors fear a trap in which they will be captured in the handover process.