Bannister left Wednesday night's game at Oakland in the sixth inning of what became a 7-2 loss after his right shoulder became stiff. Afterward, he expressed confidence that he'd be able to make his next scheduled start on Tuesday night against Cleveland.
"I think he's doing OK," Hillman said. "I don't have any news. I would hope that he would but I don't have any more than that."
After five shutout innings, Bannister's effectiveness decreased and he gave up two runs before leaving.
"I'm hoping that last night's situation was an atmospheric problem with cooler temperatures in the Bay Area and humidity in the air, but we'll see how he responds today and tomorrow," Hillman said.
Because of Bannister's situation, Hillman said he wasn't certain how the rotation would work after Monday's open date. Skipping Bannister and going directly to Gil Meche would be a possibility.
Soria was placed on the 15-day disabled list on Sunday with a stiff shoulder.
"He hasn't started throwing yet," Hillman said. "The biggest thing was to get the inflammation calmed down to where he could start some extracurricular exercises beyond what he's normally done, and we'll go from there as to where and when he plays catch."
Relievers Doug Waechter (strained left elbow) and John Bale (thyroid surgery) also remain on the disabled list.
"Waechter is here with us. He's got an oblique problem that resulted during his rehab. Bale, as of two days ago, had back-to-back days where he actually maintained his velocity and continued to regain his arm strength," Hillman said.
Dick Kaegel is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.