"The attackers, who are no doubt Boko Haram insurgents, attacked Amchaka and nearby villages this morning, hurling IEDs (improvised explosive devices) into homes and setting them on fire," Baba Shehu Gulumba, a government administrator in Bama district, told AFP.
"They then went on a shooting spree, opening fire on confused residents as they tried to flee, killing 60 people and injuring several others," Gulumba said by phone from Maiduguri, the state capital.
The attackers stormed Amchaka and neighbouring villages in Bama using trucks, motorcycles and two armoured vehicles, shooting residents and torching homes, he said.
Other local sources confirmed the attack but did not give a death toll.
The assailants vandalised boreholes, the only water source for the villagers, Gulumba added.
The attacks have prompted a mass exodus of people from villages in the area.
Scores have been killed in recent days in Borno state, according to the local lawmakers.
Fearing attack, around 400 students in the state had boycotted university entrance exams on Saturday.
The military announced a state of emergency last May in northern Nigeria and launched a major offensive to crush the Boko Haram insurgency, which has been running for five years and has claimed thousands of lives.
The rebels say they are fighting to create strict Islamic state in northern Nigeria.
Violence in the region -- which includes Yobe and Adamawa as well as Borno -- has already killed some 1,500 people this year alone.
Boko Haram, which means "Western education is sinful" in the Hausa language, has attacked isolated villages, schools and churches as well as military bases.
The conflict has forced hundreds of thousands of people to flee to other Nigerian states and neighbouring countries.
The United States has declared Boko Haram leader Abubakar Shekau a "global terrorist" and has put a $7 million bounty on his head.
Public secondary schools in Borno state were last month closed indefinitely following deadly attacks. The closure affects 85 secondary schools, catering to some 120,000 students.
In late February, 43 students were shot and hacked to death when suspected Boko Haram gunmen stormed the Federal Government College in Buni Yadi, in Yobe state.