Ted Bundy's murder kit is coming back to haunt Zak Bagans -- because Utah government officials are now dying to know if the kit actually belongs to them.

A rep for the Utah State Archives tells TMZ ... the spooky murder kit Zak recently bought could possibly be state property and IF that's the case, it was never Zak's to purchase.

What's more ... the Archives dept. has contacted the Utah State Attorney General's Office to discuss any legal options it could take to formally repossess the murder kit, and all documents and records related to the case.

TMZ broke the story ... the "Ghost Adventures" star bought a treasure trove of dangerous items seized from Bundy's infamous Volkswagen Beetle. The kit had been in former Salt Lake Co. Sheriff's detective Jerry Thompson's possession until his wife sold them to Zak.

The kit included 2 mismatched gloves, a flashlight, orange wire, Glad trash bags, a brown gym bag, white clothesline rope and strips of bed sheets used for tying up his victims.

Sources close to Zak tell TMZ ... the ghost aficionado feels he bought the collection free and clear. Plus, he feels the person he bought it from was the legal owner of the kit.

The Archives rep makes it clear ... "If these are government records, we certainly would love to have them in the custody and care of Utah State Archives."