Terms of the offer sheet were four years and $18 million, including $10 million guaranteed, a league source told ESPN NFL Insider Adam Schefter.
Already having watched quarterback Brock Osweiler (Texans), defensive end Malik Jackson (Jaguars) and linebacker Danny Trevathan (Bears) depart in free agency, the Broncos have five days to match the Dolphins' offer.
The Broncos are limited in what they can do given their pursuit of several options at quarterback, including a potential trade with the San Francisco 49ers for Colin Kaepernick and a potential free-agent visit by former Washington Redskins quarterback Robert Griffin III, and their attempt to secure a long-term deal with star linebacker Von Miller.
On Thursday, Anderson visited Miami, where former Broncos offensive coordinator Adam Gase is now the head coach. Gase knows Anderson well and has lauded the running back's potential as a full-time runner.
Several teams reportedly were interested in the 25-year-old Anderson, including the New England Patriots and the Bears. He was tendered a one-year, $1.671 million deal, with the Broncos holding right of first refusal. That means the Broncos can match any other offer, but they won't receive compensation if they don't.
Anderson made the Broncos' roster as an undrafted rookie out of California in 2013.
He was an injury replacement in the Pro Bowl following a 2014 season that saw him rush for 849 yards and eight touchdowns. He gained the bulk of those yards with his work down the stretch.
Last season, Anderson battled some injuries and was in a rotation with Ronnie Hillman. He finished the season with 720 yards on 152 carries (4.7-yard average) and five touchdowns. He had 90 yards rushing and one touchdown in the Broncos' 24-10 win over the Carolina Panthers in Super Bowl 50.
After that win, Anderson said: "I'd like to stay. We just played in the Super Bowl, we've won the division every year I've been with the team, but it's two sides. They have to want you to stay. I believe in my abilities and believe it will work out."
Hillman, who produced 863 rushing yards and seven touchdowns in 2015, is an unrestricted free agent.
Information from ESPN.com's Jeff Legwold was used in this report.