Chicago announced the agreement with Belisario on Thursday. The talks with Downs were revealed by a person familiar with the situation who spoke on the condition of anonymity because the deal had not been announced.
The 30-year-old Belisario was 5-7 with a 3.97 ERA in 77 games with the Dodgers last season and 20-12 with a 3.29 ERA over four years. He became a free agent when Los Angeles failed to offer a contract by the Dec. 2 deadline.
Belisario is scheduled to be eligible for arbitration after the 2014 and 2015 seasons.
Downs would earn $3.75 million next season, and the White Sox would hold a $4.25 million option for 2015 with a $250,000 buyout. Fox first reported his pending deal.
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