ESPN reported that the Patriots also are acquiring the Browns' fifth-round pick in the 2018 NFL Draft and will send their 2019 third-round selection to Cleveland.
The trade is the fourth by the Browns since Friday. The team traded for Miami Dolphins wide receiver Jarvis Landry, Buffalo Bills quarterback Tyrod Taylor and Green Bay Packers cornerback Damarious Randall in separate deals on Friday.
The trades can't officially be announced until Wednesday, when the 2018 league year begins.
Shelton, 24, has recorded 71 tackles and 1.5 sacks in 46 career games since being selected by Cleveland with the 12th overall pick of the 2015 NFL Draft. He had 34 tackles in 14 contests last season.
The 6-foot-2, 335-pound Shelton is slated to earn a base salary of $2.03 million in 2018.
Shelton could aid the Patriots after the team recently informed defensive tackle Alan Branch that they wouldn't be picking up his option for 2018, making him a free agent.
Branch, 33, joined New England in 2014 and was a key contributor during the 2015 and 2016 seasons, appearing in all 32 games and making 30 starts.
The 6-6, 342-pounder's production declined last season as he started six of the 12 games he played and was inactive in the postseason during the Patriots' run to Super Bowl LII.