Anderson has spent the past three seasons with the Denver Broncos and emerged as a starter despite being undrafted out of California in 2013.
The Dolphins hired Adam Gase as head coach, who is quite familiar with Anderson as serving as Denver's offensive coordinator.
The Broncos have five days to match Miami's offer.
Denver tendered Anderson a one-year, $1,671,000 deal and will not receive compensation if it doesn't match the offer.
Anderson rushed for 720 yards and five touchdowns last season. He added 90 yards and one score in the 24-10 victory over the Carolina Panthers in Super Bowl 50.
In 2014, he rushed for career highs of 849 yards and eight touchdowns.
Anderson split time with Ronnie Hillman the past two seasons. Hillman (863 rushing yards, seven touchdowns) is also a restricted free agent.
Miami targeted Anderson as a replacement for Lamar Miller, who left as a free agent and signed with the Houston Texans.
The Dolphins also announced the signings of safety Isa Abdul-Quddus, linebacker James-Michael Johnson, center/guard Jacques McClendon and tackle Sam Young, and the team said cornerback Byron Maxwell restructured his contract.
Abdul-Quddus, 26, reportedly signed a three-year, $12.75 million contract that includes a $2.5 million signing bonus and a $6 million guarantee.
Young, 28, reportedly got one year and $910,000. He received a $100,000 signing bonus and a $150,000 guarantee.