Gaffney, 25, spent the majority of the 2016 season on the Patriots' practice squad. He became expendable when New England signed running back Rex Burkhead last week.
A sixth-round draft pick out of Stanford in 2014, Gaffney has yet to appear in an NFL game.
The 6-foot, 220-pounder spent the 2014 and 2015 seasons on injured reserve. He was released by the Patriots twice last season and subsequently signed to the practice squad, spending one game on the 53-man roster.