War Machine -- Dog The Bounty Hunter On The Case ... And The Clock's Ticking

War Machine has until 7 PM Tuesday to turn himself in to cops ... or else the most famous bloodhound in the US -- Dog the Bounty Hunter -- will be gunnin' for him.

TMZ has learned ... a member of Christy Mack's family reached out to Dog through a mutual friend with MMA ties, hoping the mulletted one will succeed where the cops haven't.

Dog's told us he's already made contact with War Machine and given him the ultimatum -- 24 hours, and then you're mine. We're told War Machine was not receptive.

We're told Dog's entire crew will rock immediately after the deadline passes.

War is facing 7 criminal charges after using Christy as a punching bag during the 3-person altercation in her home Friday. She was left with 18 broken facial bones and a ruptured kidney.

Cops say he choked Christy's new BF, then licked his finger ... then rubbed Christy's genitals.