Militants in a 4-wheel drive vehicle and about five motorcycles attacked Kautikari at about 8pm local time on Monday, Ali Watika, a local resident who had fled to Chibok, a town some 10kms away, told The Anadolu Agency by phone on Wednesday.
He recalled that the insurgents first fired shots into houses to force residents out and then open fire, killing at least 13 people.
Watika said he saw corpses of the slain residents while fleeing the town on Tuesday morning.
A military source said troops are already in the town and feared the death toll could be higher.
He said the insurgents had burnt many houses and commercial centers in the town.
Boko Haram attacked Kautikari in mid-June, killing at least 21 people.
On April 14, Boko Haram militants abducted 276 girls from their school dormitories in Chibok, according to official accounts.
Boko Haram kingpin Abubakar Shekau later claimed responsibility for the abductions, offering to trade the kidnapped girls for detained militants held by the Nigerian authorities.
Since then communities near Chibok have come under heavy attacks by the insurgents.
Boko Haram raided Chibok itself in November killing at least 12 people.
For the last five years, Nigeria has battled a fierce Boko Haram insurgency that has ravaged the country's volatile northeastern region.
A seemingly emboldened Boko Haram recently stepped up its militant activity even further, seizing several areas in Adamawa, Borno and Yobe states, where it has declared a self-styled "Islamic caliphate."
Several residents have fled to Maiduguri, Borno's provincial capital, and few areas considered safer in the three states.