"It was a bad hop," Odor said. "We'll see how it feels. I'm not going to do anything today and see how it feels tomorrow."
Odor wore a metal covering over the finger on Wednesday.
"Essentially, it will be his pain tolerance as to what he can do," manager Jeff Banister said.
With Odor out, the Rangers don't have an active backup middle infielder, and they also don't have one on their 40-man roster after Adam Rosales took his free agency earlier this week. Texas traded Chris Bostick and Luis Sardinas during the offseason, and Jurickson Profar remains on the 60-day disabled list while recovering from a torn muscle in his right shoulder.
The Rangers do have veterans Ed Lucas and Tommy Fields playing at Triple-A Round Rock, and both have Major League experience, but neither is on the 40-man roster.
Corporan begins rehab assignment: Catcher Carlos Corporan, who has been on the DL with a sprained left thumb since July 13, has been sent to Round Rock on medical rehab assignment. The Rangers are hoping that he can be ready to return on Sept. 1, when the rosters can be expanded up to 40 players.
"We felt he was good enough to go out and get at-bats and get time behind the plate," Banister said. "When Sept. 1 rolls around, he could become an option -- if we need a pinch-runner or pinch-hitter, we can do that."
Catcher Robinson Chirinos, on the DL with a strained left shoulder, is not ready to join Corporan on a rehab assignment, but he is making progress and has been able to take batting practice the last couple of days without any discomfort.
"Hopefully, it will continue to improve and I can get back as soon as possible," Chirinos said. "Corporan and I have been doing everything we can to get back to the field."
Corporan could be back quicker simply because the Rangers want a third option behind Chris Gimenez and Bobby Wilson. The Rangers will likely do everything they can to prepare Chirinos to return as their No. 1 catcher.
"There are still some hurdles to cross before we get there," Banister said.
• The Rangers are planning to send first baseman Kyle Blanks to Round Rock on rehab assignment by the end of this week. Blanks has been sidelined since June 17 with inflammation in both Achilles tendons.
• Shortstop Elvis Andrus turned 27 on Wednesday.
• David Price, who started for Toronto on Wednesday, was the fourth straight left-hander the Rangers faced, but that streak is about to end. In fact, the Rangers could face nine straight right-handed starters, beginning with Toronto's Marco Estrada on Thursday.
The Orioles are scheduled to start three right-handers this weekend -- Kevin Gausman, Ubaldo Jimenez and Miguel Gonzalez -- while the Padres are expected to do the same, with Tyson Ross, Andrew Cashner and Ian Kennedy in line for the three-game series that begins on Monday in San Diego. After that the Rangers face the Angels, whose rotation is lined up for right-handers Garrett Richards and Jered Weaver scheduled before left-hander Hector Santiago.