The Tigers recalled Brieske from Triple-A Toledo on Tuesday.
Montgomery isn't in the lineup for Tuesday's game against the Astros.
The Tigers activated Greene (hamstring) from the 10-day injured list, and he's starting in left field and batting cleanup Tuesday against the Twins.
The White Sox recalled McDougal from Triple-A Charlotte on Tuesday.
The White Sox designated Gilbert for assignment Tuesday.
The White Sox selected Pham's contract from Triple-A Charlotte on Tuesday.
Montgomery went 2-for-3 with a solo home run, a double, two walks and two runs scored Monday against the Astros.
Kay (9-7) took the loss Monday, allowing six runs on six hits and two walks over 4.2 innings against the Astros. He struck out five.
Jobe (1-2) took the loss against the Twins on Monday, allowing five runs (four earned) on six hits and two walks in five innings. He struck out four.
Lee went 2-for-3 with a solo home run in Monday's 11-1 loss to the Twins.
