ESPN's Keith Law had Kay and Woods-Richardson ranked as the No. 6 and No. 7 prospects in the Mets system before the season.
MLB.com was first to report the trade.
Stroman, a first-time All-Star this season, is 6-11 in 21 starts with a 2.96 ERA. Mostly a ground ball pitcher, the 28-year-old has the third lowest home run percentage (1.