Bush is in the tail end of his career, but still has plenty of explosion left. His role with the Patriots would be very much reduced from years' past, but that is something he is likely used to after slowly losing playing time to Joique Bell in Detroit.
The Pats are still well equipped at the runningback spot with LeGarrette Blount, Jonas Gray, Brandon Bolden, James White, Tyler Gaffney and Dion Lewis. Signing Bush isn't a necessity, but it sure would help. Otherwise it may be Brandon Bolden or James White who fall into the third down back role.
Bush is coming off a lukewarm year (76 carries, 297 yards). He struggled with injuries, playing in only 11 games. It's the the most time he's missed in his career since 2010. It's possible he may be wearing down after getting over 200 carries from 2011-13. Still, playing a bit part on a championship caliber team is a role he's most certainly capable of playing. Especially in the hands of Bill Belichick, who always tries to put players in the best position to succeed.