The team doesn't have a clear timeline for Tanaka's return, but general manager Brian Cashman estimated that he will be out a "month" after he was asked for a timetable multiple times by reporters Saturday at the ballpark.
"It's very possible he's back before [the All-Star Break]," manager Aaron Boone added.
Neither Boone nor Cashman provided specifics regarding the changes the Yankees will make to their rotation. Cashman said there was an "internal option" the organization has been discussing, but he didn't say who. That player is likely at the minor league level.
"We have an idea of what we're going to do," Cashman said. "We're not going to deal with it now because things can change injury-wise or what have you."
With a day off Monday, Boone estimated that it wouldn't be until Friday that the Yankees need to address the vacancy in the rotation.
"We feel like we're covered still," he said.
At Triple-A, 22-year-old left-hander Justus Sheffield appears the most likely candidate to get a call-up. Like Tanaka, he pitched Friday night. In a six-inning performance for Scranton/Wilkes-Barre, he allowed two unearned runs, a hit and three walks. He also struck out eight, with his fastball hovering around 95-96 mph.
Sheffield entered the season as the 16th-rated prospect in Keith Law's Top 100 for ESPN.
Jonathan Loaisiga, a 23-year-old right-hander currently pitching for Double-A Trenton, is another candidate.
Tanaka's injury came one day after fellow Yankees starter Jordan Montgomery underwent season-ending Tommy John surgery. Even before that surgery, the Yankees likely were in the market for a pitcher ahead of the trade deadline.
"We're always open to outside help, no matter what time of year it happens to be," Cashman said. "But we also have internal options that we can rely on."
Cashman highlighted the recovery of another injured pitcher, Luis Cessa (oblique). The right-hander could eventually give the Yankees added starting depth depending upon his progression. Per Cashman, a second rehab start is coming up in the next few days for Cessa.
Tanaka was hurt sprinting from third base to home in the sixth inning of Friday's win over the New York Mets at Citi Field.
"It's not something that I normally do or that we pitchers in the American League normally do," Tanaka said of running the bases. "But that's part of baseball. It's on me that I got hurt. I don't think it's about the rules."
When Judge lofted a fly ball to right field, Yankees third base coach Phil Nevin got Tanaka positioned back on the third base bag, tagging up. As soon as Mets right fielder Jay Bruce made catch, Tanaka was sent home. Bruce's throw was a little wide, causing catcher Devin Mesoraco to vacate the plate just long enough for Tanaka to score without sliding.
As he walked back to the Yankees dugout, Tanaka eased in noticeably ginger. A half-inning later, he was replaced by reliever Jonathan Holder.
The injury derailed what had been a strong night for Tanaka on the mound. He lasted five innings, allowing just one run and one hit -- a leadoff home run to Brandon Nimmo. He also struck out eight batters.
On the season overall, Tanaka is 7-2 with a 4.58 ERA in 13 starts.
In the corresponding roster move Saturday, the Yankees recalled infielder Ronald Torreyes from Triple-A.