The AL Central leaders purchased Purke's contract before Friday night's game at Yankee Stadium.
The 25-year-old Purke hasn't played in the majors. He was 0-0 with a 2.30 ERA in 10 games at Charlotte, including one start. The White Sox signed him in November after he spent four seasons in the Washington system.
Purke takes the roster spot of righty Scott Carroll, who was optioned to Charlotte on Thursday.
The 31-year-old Danks was designated for assignment on May 5. He was 0-4 with a 7.25 ERA in four starts for Chicago. Danks was 79-104 with a 4.38 ERA in 10 seasons with the White Sox.
The White Sox are responsible for the remaining portion of Danks' $14.25 million salary this year, a little more than $11 million.