The trade is pending physicals.
Britton, 30, has a 3.45 ERA and four saves in 16 appearances this season. He had a 2.89 ERA and 15 saves in 38 appearances during the 2017 season. The two-time All-Star led the American League with 47 saves in 2016. He also led the league in games finished in 2015 and 2016.
The 6-foot-3, 195-pound southpaw is due $12 million in 2018, the final year of his contract. The Yankees will pick of the remainder of Britton's salary this season -- about $5 million -- according to MLB.com.
Tate is the Yankees' No. 9 prospect while Carroll is ranked No. 15 in the Yankees' system. Tate, 24, was the No. 4 overall pick in the 2015 MLB June Amateur Draft. He has a 3.38 ERA and a 5-2 record in 15 starts in Double-A this season. Carroll, 25, had a 2.38 ERA and nine saves in 32 games in Triple-A this season. Rogers, 24, had a 3.95 ERA and a 6-8 record in 19 starts this season in Triple-A.