The White Sox signed Wilson to a one-year, $1.05 million contract Wednesday, Ken Rosenthal of The Athletic reports.

Wilson elected free agency Monday, Adam McCalvy of MLB.com reports.

Wilson (oblique) could be activated from the 15-day injured list as soon as Wednesday, Adam McCalvy of MLB.com reports.

The Brewers placed Wilson on the 15-day injured list Tuesday with a right oblique strain, Adam McCalvy of MLB.com reports.

Manager Pat Murphy said Thursday that Wilson will pitch out of the bullpen going forward, Curt Hogg of the Milwaukee Journal Sentinel reports.

Wilson will be used as a bulk reliever in Monday's game versus the Rockies, Adam McCalvy of MLB.com reports.

Wilson (5-3) picked up the win Tuesday against the Rangers, allowing three hits with no walks across six scoreless innings. He struck out four.

Wilson will pitch behind opener Hoby Milner against the Rangers on Tuesday, Adam McCalvy of MLB.com reports.

Wilson did not factor into the decision in Thursday's loss to the Padres, allowing five runs on seven hits and two walks over 4.2 innings while striking out five.

Wilson is in line to serve as the Brewers' bulk reliever Saturday versus the Reds, Curt Hogg of the Milwaukee Journal Sentinel reports.