"Both players will undergo further diagnostic testing tomorrow and we will provide an update during tomorrow night's game," Twins general manager Bill Smith said in a statement.
Gibson was originally placed on the Minor League disabled list with soreness in the flexor muscle area near his elbow and underwent an MRI on Wednesday. The 6-foot-5 right-hander is the 33rd-ranked prospect in baseball and No. 2 in the Twins' system, according to MLB.com.
The 23-year-old could need Tommy John surgery, as Twins Minor League director Jim Rantz said last week the club was worried it could be a torn ulnar collateral ligament.
Gibson, the club's first-round pick in the 2009 First-Year Player Draft, had been struggling recently, as he went 0-5 with a 6.47 ERA in eight starts over June and July. It came after he entered June with a 3.60 ERA and 59 strikeouts in 55 innings.
Slama, meanwhile, was placed on the Minor League disabled list on July 6 because of elbow soreness. The reliever posted a 2.92 ERA with 42 strikeouts in 37 innings with Triple-A Rochester before his injury. The 26-year-old also tossed 2 1/3 scoreless innings with the Twins this year in a brief stint with the club from late May into early June.