The Dolphins, who entered free agency with the league's most salary-cap space, get a player, in Van Noy, who is a favorite of coach Brian Flores from their time together in New England. Van Noy's versatility will be used at linebacker and as an edge rusher.
Van Noy, who turns 29 on March 26, was an unrestricted free agent for the first time in his NFL career.
After entering the league as a second-round draft choice with the Detroit Lions in 2014, his career was revived when the Patriots acquired him and a seventh-round pick from the Detroit Lions in exchange for a sixth-round pick in October 2016.
Having fallen out of favor in the Lions' 4-3 defense, Van Noy became a key cog on two Super Bowl championship teams. His versatility to play multiple linebacker spots and rush the passer made him an ideal fit in New England's defense.
In three-and-a-half seasons, Van Noy played in 51 regular-season games and totaled 250 tackles, with 16.5 sacks, 4 forced fumbles (4 recoveries), 2 interceptions and 3 defensive touchdowns. He also appeared in 10 playoff games, adding 46 tackles with 5.5 sacks and 4 forced fumbles.
The Patriots signed him to a two-year, $11.5 million extension in September 2017.
ESPN's Mike Reiss contributed to this report.