The Patriots are also acquiring the Browns' 2018 fifth-round pick and will send their 2019 third-round pick to Cleveland, sources told ESPN.
The trade, which can't officially be announced until Wednesday, when the 2018 league year begins, is the fourth trade the Browns have agreed to since Friday.
On Friday, the Browns traded for Miami Dolphins wide receiver Jarvis Landry, Buffalo Bills quarterback Tyrod Taylor and Green Bay Packers cornerback Damarious Randall in separate deals. Shelton joins quarterback DeShone Kizer, traded to Green Bay in the Randall trade, as players dealt away by the Browns.
The 6-foot-2, 335-pound Shelton entered the NFL as a first-round draft choice of the Browns in 2015 (No. 12 overall).
He has played in 46 regular-season games (45 starts), totaling 71 tackles and 1.5 sacks. The Washington alum was a highly touted prospect coming out of college, in part because of his strength at the line of scrimmage against the run.
Shelton, 24, is scheduled to earn a base salary of $2.03 million in 2018. The Patriots can also pick up his fifth-year option for 2019, which would be a yet-to-be-determined salary of the average of the third through 25th highest salaries among defensive linemen.
Shelton joins 2015 first-round pick Malcom Brown and seven-year veteran Lawrence Guy atop the Patriots' interior defensive line depth chart. The team also has 2016 third-round pick Vincent Valentine, who spent all of 2017 on injured reserve, at the position, along with 2017 undrafted free agent Adam Butler, who is more of an interior sub rusher.
In Cleveland, Shelton has played under three defensive coordinators -- Jim O'Neil (2015), Ray Horton (2016) and Gregg Williams (2017). O'Neil and Horton ran a 3-4 defense that relies on a strong nose tackle, but with the Browns switching to a 4-3 under Williams in 2017, Shelton might not have had the same value to them.
In New England, which runs a multiple defense that often asks defensive linemen to two-gap and play the run first, Shelton's skill set should fit well.
Shelton helps fill a void after the Patriots recently informed veteran defensive tackle Alan Branch that they wouldn't be picking up his option for 2018, making him a free agent, and veteran defensive tackle Ricky Jean Francois said Friday that he plans to play elsewhere in 2018.
ESPN's Adam Schefter and Mike Reiss contributed to this report.