The FBI has reached out to authorities in Mexico, in the hunt for Nancy Guthrie and her kidnapper, TMZ has learned … and TMZ has received a new ransom note.

Law enforcement sources involved in the investigation tell us … the FBI has contacted Mexican federal law enforcement to spread the word to various police agencies. We’re told so far there have been no solid leads.

Our sources say the FBI believes it’s possible Nancy was taken across the border, but it’s unlikely it happened directly after the kidnapping. Authorities have checked Border Patrol cameras and other electronic devices, and they came up dry. That said, our sources say it’s possible Nancy could have been taken across the border at some other time. Of course, there are ways to evade authorities when crossing the border.

TMZ has obtained another ransom note — this one a highly sophisticated demand involving a cryptocurrency other than bitcoin. The email demands a dollar amount similar to the $6 million ransom demand in the previous email we received, days after the kidnapping. This email graphically describes the consequences if the ransom isn’t paid.

One thing the FBI wants to make clear … anyone who tries to scam the Guthrie family by making fake ransom demands is playing with fire, because, as one source puts it, “They will go to federal prison for a long, long time.” Based on what we know, this is not a hollow promise.

A few other items … law enforcement sources say they did not get any leads from the gun stores the FBI contacted. As we reported, the FBI created a list of people who purchased the type of holster the kidnapper was wearing and presented that to various gun store owners in the Tucson area, asking if anyone with those names bought a gun in the last year. No luck.

We’re also told there is an FBI team working with Walmart and other retailers to find who may have purchased some of the items the kidnapper was wearing. Again, no breakthroughs so far.