The Broncos also sent a seventh-round pick (No. 225) in next month's draft to Minnesota and obtained a fifth-round selection in the 2019 draft from the Vikings.
Siemian was Denver's seventh-round pick in 2015 from Northwestern and started 24 of 26 games.
Siemian, 26, was named the starting quarterback out of training camp each of the past two seasons.
Last season, Siemian started 10 of the 11 games he appeared in and completed 206 of 349 passes for 2,285 yards, with 12 touchdowns and 14 interceptions for a quarterback rating of 73.3.
The Broncos were 5-5 in the 10 games he started in 2017 and went 8-6 in his 14 starts in 2016.
In three seasons, Siemian has completed 59.3 percent (495 of 835) of his passes for 5,686 yards with 30 touchdowns and 23 interceptions for a quarterback rating of 79.9.
With Minnesota, Siemian is expected to be a backup for recently acquired Kirk Cousins.
The Broncos signed free-agent quarterback Case Keenum to be their starter next season.
Paxton Lynch, a first-round draft pick in 2016, has started two games each of the past two seasons for Denver and is the Broncos' backup quarterback at the moment.