The FBI and the Pima County Sheriff’s Department are homing in on the outfit the presumed kidnapper wore as he approached Nancy Guthrie‘s front door, and they are on the hunt for who bought the items … law enforcement sources involved in the investigation tell TMZ.

Our sources singled out 3 items — the gloves, the backpack, and the gun visible on the Nest video.

We’re told authorities are contacting manufacturers and retailers to figure out who may have purchased the items.

BTW … we’ve determined the holster in question appears to be polyester, which is typically used for carrying a revolver … but in this case the weapon looks like a semi-automatic handgun.

The backpack is very similar to one sold by Walmart — the front straps of Walmart’s Ozark Trail Hiker backpack have the same reflective pieces as the one visible in the video.

As TMZ reported, authorities believe the person who wrote the ransom notes is from the Tucson, Arizona area. They have various reasons why they believe that, but a big one is that both ransom notes were sent to local TV stations.

Even if the person who purchased the items paid cash, big stores like Home Depot could pinpoint date and time of sale, and then check surveillance video that would show someone in the store and at checkout.