Sreesanth hurt his left calf during the first one-day international last week and has been replaced by left-arm seamer Rudra Pratap Singh, who was also a member of the one-day squad.
"Despite the best efforts at treatment of his injury he will not be 100 percent fit for the start of the first Test ... and he will continue to require treatment for this soft tissue injury for another seven days," a statement said.
Sreesanth, 24, is one of India's new crop of pacemen and has impressed with his spirit and wicket-taking abilities.
The first Test in Chittagong begins on Friday while the second and final Test will be held in Dhaka from May 25.
India won the three-match one-day series 2-0 after the final game was abandoned on Tuesday because of a waterlogged outfield.