Burke will serve as the Angels' opening pitcher in Thursday's game versus the Blue Jays, Erica Weston of Bally Sports West reports.
The Angels claimed Burke off waivers from the Rangers on Tuesday.
The Rangers designated Burke for assignment Sunday.
The Rangers recalled Burke from Triple-A Round Rock on Sunday.
The Rangers optioned Burke to Triple-A Baltimore on Sunday, Evan Grant of The Dallas Morning News reports.
Burke (hand) will begin a rehab assignment in the Arizona Complex League on Thursday.
Burke (hand) was transferred to the 60-day injured list by the Rangers on Thursday.
The Rangers placed Burke on the 15-day injured list Saturday due to a broken right hand, Jeff Wilson of RangersToday.com reports.
The Rangers added Burke to their World Series roster Tuesday, Kennedi Landry of MLB.com reports.
Burke wasn't included on the Rangers' roster for the ALCS against the Astros, which starts Sunday night in Houston, Evan Grant of The Dallas Morning News reports.