The deal is worth $1.5 million that includes incentives that could bring the total value of the deal to a max of $3.5 million, a league source told ESPN's Adam Schefter.
Foster's agent, Mike McCartney, tweeted congratulations to the former Houston Texans star on Monday afternoon.
Foster will join a Dolphins backfield that is in flux after Lamar Miller, Miami's leading rusher last season, signed with the Texans. Jay Ajayi and rookie Kenyan Drake currently are atop the Dolphins' depth chart at running back.
The Texans did not offer Foster a pay cut before releasing him on March 3. They couldn't count on his availability anymore, as he approaches his 30th birthday next month.
The Texans were set to spend nearly $9 million in cap space on Foster's salary, but Texans general manager Rick Smith said the decision to release Foster would not be based on his salary.
Foster only played in four games last season, hampered by two significant injuries. He suffered a torn groin in training camp that caused him to miss three games. He returned for four games, but then suffered a torn Achilles tendon in the final five minutes against the Dolphins.
Since becoming the Texans' starter in 2010, Foster has had a torn meniscus, multiple hamstring injuries, a knee injury, a back injury that required surgery, a minor groin injury, a major groin injury and a torn Achilles tendon. In that time, Foster has carried the ball an average of 20 times per game, the highest average in the NFL.
Foster came to the Texans as an undrafted free agent in 2009. He spent most of that season on the practice squad, being elevated to the active roster in November.
The 2010 season marked his breakout year. Foster rushed 33 times for 231 yards in a season-opening win over the Indianapolis Colts, sealing his role as the Texans' starting running back. He won the rushing title that year, gaining 1,616 yards on 327 carries, and went to his first of four Pro Bowls after that season.
ESPN's Adam Schefter and Tania Ganguli contributed to this report.