Burke did not factor into the decision in Sunday's 4-2 loss to the Padres, allowing one run on two hits and a walk while striking out eight over six innings.
Burke is scheduled to start Sunday's game against the Padres in San Diego.
Burke (1-0) yielded two runs on five hits and two walks over five innings Sunday, striking out five and earning a win against the Athletics.
Burke is listed as the White Sox's probable starting pitcher for Sunday's game against the Athletics in Chicago.
White Sox interim manager Grady Sizemore said Wednesday that Burke is "in line" to get a start the next time the club has an opening in the rotation, Vinnie Duber of AllCHGO.com reports.
Burke struck out three batters and allowed one unearned run on three hits and one walk over three innings of relief in Tuesday's 5-0 loss to the Guardians.
The White Sox selected Burke's contract from Triple-A Charlotte on Tuesday.
Burke (shoulder) should rejoin the Triple-A Charlotte rotation in May, Jim Callis of MLB.com reports.
Burke is on the 7-day injured list at Triple-A Charlotte due to shoulder soreness and is expected back sometime this month, Jim Callis of MLB.com reports.