An advertisement running on local television and radio stations warned people in Zamfara, Katsina and the Birnin Gwari area of Kaduna state to leave the forests in advance of a "heavy bombardment."
The advertisement said the bombardment would "protect the life and property of Nigerians."
A military official reached by phone confirmed the advertisement's authenticity but declined to share his name.
Murtala Alhasan Umaru, general manager at Zamfara state TV and radio, showed the advertisement to Reuters and said the military asked them to play it. There are versions in local pidgin English, Hausa, Kanuri and Fulani.
One man in Zamfara state, Abdullahi Abubakar, said he had heard the broadcast on the radio and had seen military fighter jets flying overhead.
Two other residents in Zamfara, one in Gumi and the other in Shinkafi, said there had been daily bombings since Saturday morning. The two asked Reuters not to identify them by name.
Armed gangs of men, known locally as bandits, have killed and kidnapped hundreds across northwest Nigeria over the past two years, typically operating from remote forests. The country's thinly stretched armed forces have struggled to secure the large, remote regions.