Griffin was placed on the 10-day disabled list on Saturday after leaving Friday night's 7-6 loss to the Blue Jays in the second inning. He was diagnosed with a left intercostal strain and will undergo an MRI when the Rangers return to Texas for an eight-game homestand beginning Monday.
With the Rangers' current rotation order, Griffin's spot will come back up Wednesday for the series finale against the Rays.
Texas entered play Saturday with a nine-man bullpen, but after throwing 84 pitches in relief Friday and 178 over the past seven days, long man Austin Bibens-Dirkx will need some time to bounce back.
"He'll need a couple of days, for sure," Banister said. "Probably three."
Depending on the Rangers' bullpen needs in the coming days, Bibens-Dirks or Dillon Gee, who was recalled in place of Griffin, could conceivably line up to make the Wednesday start. Gee is penciled into the bullpen for the time being, but has started nine games for Triple-A Round Rock, posting a 3.88 ERA. He last pitched on Wednesday, throwing six innings and striking out nine.
Right-hander Tyson Ross is progressing through his rehab well, but Banister said the right-hander will continue to focus on his recovery and won't be pushed to line up with the rotation spot.
"That's more important," Banister said. "That's more long-term [thinking] than short-term."
Ross is still scheduled to pitch in Triple-A on Monday against Omaha, with expected limits of six innings and 90 pitches.
Elsewhere on the Texas 40-man roster, knuckleballer Eddie Gamboa could present another option but has struggled through his first nine Triple-A starts, posting a 7.09 ERA. Fellow right-hander Connor Sadzeck has fared slightly better at the Double-A level. The big 25-year-old holds a 4.75 ERA over 10 starts, while striking out 63 batters in 53 innings.
• Texas made a roster move just after the start of Saturday's game, claiming 1B/OF Peter O'Brien from Cincinnati. The Rangers will initially assign O'Brien to Double-A Frisco, where he'll play Sunday and Monday before being sent to Triple-A Round Rock effective Thursday. Tyson Ross was transferred from the 10-day DL to the 60-day DL to make room on the 40-man roster. O'Brien had recently been designated for assignment by the Reds. In 32 games at the Triple-A level this season, the 26-year-old is hitting .168 with four home runs.
Keegan Matheson is a reporter for MLB.com based in Toronto. This story was not subject to the approval of Major League Baseball or its clubs.