The 6-foot-3, 282-pound Smith has spent his entire career with the Saints (2004-2013) and was released Feb. 12, in part because of salary-cap considerations. He was due $11.55 million in salary and bonuses in the final year of his contract.
Smith, who ranks fourth in Saints history with 67.5 career sacks, missed last season because of a torn ACL.
The 32-year-old Smith would likely project to an Andre Carter-type nickel-rushing role in New England, where ends Chandler Jones and Rob Ninkovich are locked in as starters. The Patriots' need at the position was highlighted Monday, as there is a notable drop-off after Jones and Ninkovich.