Kuroda was injured Wednesday night against Boston when Shane Victorino's line drive deflected off his pitching hand. The right-hander remained in the game for a few more batters but struggled with his control and was lifted with one out in the second inning.
The 38-year-old Kuroda said a trainer advised him not to throw Thursday, but he will attempt to complete his regular bullpen session Friday. Until then, he won't know whether he can take his next turn in the rotation.
Yankees manager Joe Girardi said Kuroda remains scheduled to start Monday in Cleveland.
Kuroda's outing in a 7-4 loss to the Red Sox was the shortest of his six-year major league career. X-rays on the pitcher's finger and a CT scan were negative.
Kuroda was the team's most reliable starter last season, going 16-11 with a 3.32 ERA in 33 starts and 219 2-3 innings during his first year with New York.
The banged-up Yankees already have five All-Stars on the disabled list: shortstop Derek Jeter, third baseman Alex Rodriguez, first baseman Mark Teixeira, outfielder Curtis Granderson and pitcher Phil Hughes.