Kluber is expected to miss at least two months because of a right shoulder injury, while Voit will miss time due to a right oblique strain.
Yankees manager Aaron Boone announced the news shortly after the Yankees' scheduled game against the Toronto Blue Jays was postponed due to inclement weather. The contest will be made up as part of a doubleheader Thursday.
Kluber was pulled from his start Tuesday against the Blue Jays after just three innings. A subsequent MRI on his injured shoulder revealed a subscapularis strain, which is expected to keep the veteran pitcher from throwing for at least four weeks.
The 35-year-old Kluber then will undergo a rehabilitation process that could last another four weeks.
"It's tough news to get, considering how well he's throwing the ball and all he's been through to get back," Boone said. "Hopefully we can start the recovery process for him and have him back in our rotation sooner rather than later."
Kluber also is being sent for a second MRI with dyeing, a more invasive exam to check whether there is additional damage.
The two-time American League Cy Young Award winner threw 58 pitches in the outing Tuesday. He allowed two runs, striking out five and walking three.
Kluber has a 4-3 record and 3.04 ERA this season.
Voit started to experience discomfort in his side Saturday that worsened Tuesday, leading to an MRI that revealed a Grade 2 strain.
Voit, who led MLB with 22 home runs last season, has a .182 batting average with one homer in 12 games after missing the start of the season following knee surgery in the spring.