Karen Klein -- Student Targeted in Hostage Hoax

Cops believed one of the students who taunted bus monitor Karen Klein was at the center of a serious hostage situation yesterday ... but it turned out to be a hoax.

Steve Chatterton -- Captain of the Greece Police Dept. in Greece, NY -- tells TMZ ... someone called the cops yesterday claiming to be holding a family hostage at gunpoint.

The caller gave police an address ... which turned out to be a home where one of the kids on Karen's bus lived ... so cops reacted with full force.

We're told cops raced to the location ... set up a perimeter ... and tried to make contact with the family, which proved difficult because the fam had turned off their phones after receiving TONS of harassing phone calls.

But cops eventually spoke with family members and realized there was no danger ... and determined the whole thing was a hoax.

Still, Captain Chatterton tells us the threats are coming in fast and furiously ... targeting the bullies and their siblings, parents, girlfriends and boyfriends. Chatterton says students who weren't even on the bus are getting threats.