Details of the violent fighting in the Monguno area of Borno state near Lake Chad are sketchy because of lack of mobile communications in the north-east.
It was sparked after so-called Islamic State (IS) announced last month that Abu Musab al-Barnawi was the leader of Boko Haram, replacing Abubakar Shekau, the reports claim.
However, Mr Shekau, who initially pledged the group’s allegiance to IS, said he was still in charge.
Some analysts believe the split could help the Nigerian military in its efforts to defeat the insurgents.
There has been no comment yet from the Nigerian authorities on the fighting.