The source said the army's Kaduna-based One Division was leading the operation and "will soon have the bandits in their sights".
The soldiers were backed by the local police, intelligence agency and air force, as well as the Kaduna state vigilance service, a vigilante group that knows the local terrain, the source added.
"The security agencies and the state government are working tirelessly to ensure the freedom of all the abducted students and pupils. We are making progress," said Muhammad Shehu Lawal, a spokesperson for Kaduna state governor, without giving details.
The Nigerian army did not respond to requests for comment.
The mass kidnapping last Thursday, the first since July 2021, shattered the dusty town of Kuriga, 90 km from Kaduna state capital, with parents waiting for answers from authorities.
Kidnappings at schools in Nigeria were first carried out by jihadist group Boko Haram, who seized more than 200 students from a girls' school in Chibok in Borno state a decade ago. Some of the girls have never been released.
But the tactic has since been adopted by criminal gangs without any ideological affiliation seeking ransom payments, with authorities seemingly powerless to stop them.
The kidnappings are tearing apart families and communities who have to pool their meagre savings to pay the ransoms, often forcing parents to sell their most prized possessions like land, cattle and grains to secure their children's release.
In Nigeria, news of kidnappings often fades quickly as abductions have become an almost daily occurrence. One of the top items on news bulletins on Monday was President Bola Tinubu visiting northwestern Niger state, where an airport was being renamed after him.
But in Kuriga, anxious parents were growing weary.
Bala Ibrahim, whose son is among the missing children, said there was no update from local authorities on the whereabouts of the pupils.
"The only thing we know for sure has happened since the abduction is that soldiers have been deployed and have blocked all routes linking Birnin-Gwari and Zamfara (state) forest," Ibrahim told Reuters by phone.
"The soldiers are in the bush going after the kidnappers."
According to Lagos-based consultancy SBM Intelligence 4,500 people have been kidnapped throughout Nigeria since Tinubu took office last May.
Security analysts say the president, who has promised to tackle widespread insecurity, has yet to lay out a clear policy on how he intends to make Nigeria safer.
"This may continue for much of the year," said Ikemesit Effiong, partner and head of research at SBM Intelligence.
