Puk, 24, ranks as the No. 43 prospect in baseball and the fifth-best left-handed pitcher, according to MLB Pipeline.
The sixth overall pick in the 2016 MLB Draft will reportedly work out of the bullpen for the A's, Gallegos reports.
Puk missed the entire 2018 season after requiring Tommy John surgery and began throwing in games again in early June. Since then, he's authored a 4.97 ERA in 25 1/3 innings across three minor-league levels, while striking out 38 of the 104 batters he's faced.
The lefty will become the second-highest selection of the 2016 class to make the majors, trailing only Cincinnati Reds outfielder Nick Senzel, who was taken second overall.