The Patriots are reportedly set to add a veteran tight end in advance of training camp.
The 26-year-old Byham caught three passes for 38 yards in four games for the Buccaneers in 2013. He was placed on injured reserve in October after sustaining calf and knee injuries.
Byham began his career with the 49ers, who selected him in Round Six in 2010. He signed with Tampa Bay in October 2012.
Byham’s addition would give the Patriots six tight ends. The Pats’ top tight end, Rob Gronkowski, could be in for a light workload this summer as he makes his way back from a torn ACL suffered last December.