Both Garcia and manager Joe Girardi are optimistic that the veteran right-hander can return to the mound during the Yankees' four-game series in Minnesota, which starts on Thursday.
On Sunday, Girardi ruled out a trip to the disabled list for Garcia because his cut wasn't "that serious."
"I think it's realistic Freddy might be starting in Minnesota, I do," Girardi said. "I don't see any reason to put him on the DL."
With Sunday's rainout, A.J. Burnett, who was scheduled to start in place of Garcia Sunday, will now pitch Monday in Kansas City. Ivan Nova will start Tuesday and Bartolo Colon will get the ball on Wednesday. CC Sabathia is scheduled to start Thursday, when the Yankees open a four-game series against the Twins.
When Garcia returns, the Yankees can either continue to use a six-man rotation, or, if they choose to go with a traditional five-man rotation, they will presumably have to remove one starter.
For the past two weeks, the Yankees had been using a six-man rotation. Entering the weekend series against Tampa, Girardi said he would pare the rotation down to five by Monday.
Garcia's injury allowed the Yankees to avoid demoting one of their starters.
But if the Yankees decide to go with a five-man rotation when Garcia returns, the decision will likely come down to demoting either Phil Hughes or Burnett.
And it will be difficult to remove Hughes from the rotation after the way he pitched Saturday, allowing two earned runs in six innings in a win over Tampa.
Burnett, on the other hand, hasn't won since June 29. He is 0-3 with a 6.00 ERA in that span, and has continued to be a lightning rod for Yankees fans. But general manager Brian Cashman launched into an impassioned defense of Burnett before Friday's game against Tampa.
"We got six guys that are capable of pitching in a rotation in a playoff race and that's a good thing, so somebody has to go and we're going to make that decision, but this stuff about (how) A.J. Burnett is worthy of being ripped from the rotation is a bunch of crap," Cashman said.
There has never been any public discussion of taking Garcia out of the rotation.
Signed to a minor league contract in spring training, the 35-year-old journeyman has a 10-7 record and 3.16 ERA through 20 starts in 2011.
Garcia said on Sunday that he cut his finger accidentally with a knife "like five days ago." He got through a bullpen session without any issues earlier in the week but had trouble throwing his split-fingered fastball on Saturday and was scratched from his Sunday start.
"I cannot go out there and not throw my best pitch," Garcia said on Sunday. "Now we have to wait (for) it to heal and go from there."
Girardi noted that the team would wait for the cut to heal completely before Garcia started throwing. From there, he would have to throw a bullpen without incident before returning to the mound.
Girardi has not announced a starter beyond Thursday, leaving open the possibility that either Hughes or a healthy Garcia could start on Friday against the Twins.
"I'm hoping Freddy can go in Minnesota, I'm not sure yet," Girardi said. "So I still don't have to do anything there."
Ian Begley is a regular contributor to ESPNNewYork.com.