Morosi tweeted that talks between the Astros and the Pirates were "imminent".
There has been no word yet on what the Pirates will receive from the Astros.
Gerrit Cole was drafted by the Pirates with the first overall pick in 2011, and spent the last 5 years as a starter in Pittsburgh.
Cole will strengthen one of the top staffs in the majors, as the Astros now employ Justin Verlander, Dallas Keuchel (foot) and the former Pirate. In addition, Cole has allowed 749 hits and 327 runs allowed. His best season came in 2015 when he went 19-8 with a 2.60 ERA, 1.091 WHIP and 202 strikeouts in 32 starts.
In 2013, specifically in September, Cole was named the NL Rookie of the Month.