ESPN was first to report the trade that also includes the Browns dealing a 2018 fifth-round pick and receiving a 2019 third-round pick in return.
Shelton enters 2018 on the final year of his rookie contract but since he was a first-round pick, the Patriots have a fifth-year option.
Shelton joins a defensive tackle group led by fellow 2015 first-round pick Malcolm Brown, along with fellow tackles Lawrence Guy and Vincent Valentine.
On Friday, Dorsey acquired quarterback Tyrod Taylor, wide receiver Jarvis Landry and cornerback Damarious Randall.
The trades can't officially be announced until Wednesday, when the 2018 league year begins.More news: Russian spy attack: British troops deployed to assist probe
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For the fourth time in less than 24 hours, the Cleveland Browns have pulled off a trade.
By adding Shelton, the Patriots not only add a once promising prospect to their defense, but get younger. The 6-foot-2, 335-pounder, who starred in college at Washington, started 45 of 46 games in three seasons with Cleveland and the team seemed pleased with his progress against the run.
In 2017, Cleveland moved to a 4-3 system and Shelton's struggles continued. Considering that the Patriots only gave up a 2019 third round pick, getting a player the caliber of Shelton will certainly help. The former Brown appeared in all but two games throughout his career, an impressive feat for a position as physically demanding as defensive tackle.
Shelton specializes as a run-stuffing defensive tackle.
Shelton will fill in nicely for the Patriots on early downs, as he should be a good replacement for Alan Branch.