ESPN reported the pick is a 2017 conditional seventh-round selection.
Bostic played in 11 games, including one start, for the Patriots last season. Before that, he played two seasons with the Chicago Bears.
He was a second-round selection by the Bears in the 2013 NFL Draft .
The Patriots also signed defensive tackle Anthony Johnson, who played in seven games with the Miami Dolphins last season, but did not start any of them. He was originally signed as an undrafted free agent in 2014.