Nova, 31, has one year remaining on his current three-year $26 million deal and is owed $9.1 million in 2019.
Nova went 9-9 with a 4.19 ERA for the Pirates last season and carries a 78-64 overall record over nine seasons, a career that started with the New York Yankees. The 6-foot-5 right-hander was the Pirates' opening day starter in 2018.
"Ivan is a quality individual who provides a veteran presence to our starting rotation and has shown the ability to consistently throw strikes," White Sox general manager Rick Hahn said in a statement released Tuesday.
Rosario, 19, went 1-4 with a 2.57 ERA and 70 strikeouts in 14 games (11 starts) between the Dominican Summer League and Arizona Rookie League.
The Pirates also received a $500,000 increase in their international bonus pool for investing in this year's market.
--Field Level Media