Spinner R Ashwin and pacemen Ishant Sharma, Bhuvneshwar Kumar and Umesh Yadav have all been rested, the Board of Control for Cricket in India announced on its website.
Rasool, 24, has taken 46 wickets in 17 first-class matches and has proven himself handy with the bat, making three first-class centuries and boasting an average of 38.57.
Mohit Sharma, also 24, has claimed 44 wickets in his 11 first-class outings and has a batting average of 37.31.
Batsman Virat Kohli will continue as captain in the absence of Mahendra Singh Dhoni (hamstring injury) for the tour, which comprises five one-day internationals.
The first match will be played in Harare on July 24.
India squad: V Kohli (captain), S Dhawan, Rohit Sharma, D Karthik, C Pujara, S Raina, A Rayudu, A Rahane, R Jadeja, A Mishra, P Rasool, M Shami, V Kumar, J Unadkat, Mohit Sharma.