Savannah Guthrie and her family say they will pay the ransom for the return of their mother, Nancy.

The NBC anchor and her family released a video Saturday afternoon in which they directly address the people allegedly holding their mother captive ... saying they received their message, and they will pay the demanded dollar figure.

Nancy was last seen when a family member dropped her off at her Tucson home on Saturday, January 31. We and several other news outlets received an alleged ransom note earlier this week which contained two deadlines.

The first deadline was Thursday at 5 PM which changed the alleged abductors demands. The second deadline is Monday, February 9.

Guthrie and her family previously asked for proof of life before they paid a ransom. It is unclear if they received such confirmation ... but, this second video seems to indicate they feel these abductors are serious.

On Friday, our own Harvey Levin broke down how far Nancy may have been taken showing a geographic area where her abductors may be holding her based on clues in the alleged ransom note we received.

As we've told you ... Nancy is 84 years old and not in terrific health. She has a pacemaker and needs daily medication. It's unclear if she's receiving her pills.

Law enforcement officials have made it clear they have not identified a suspect, person of interest or vehicle connected to the crime. However, a man was arrested for allegedly sending fake ransom demands to the Guthries.