Bumrah was selected in India's 15-man group but will need to first be cleared after suffering niggle on tour in England, while batter Sai Sudharsan is in the boat.
Ravindra Jadeja has recovered from tennis elbow and is part of the squad but Washington Sundar is unavailable for selection for the first Test with a hamstring injury.
The 33-year-old Jain, an off-spinner who can also bat in the top 7, impressed for India A on the recent to Sri Lanka where he scored an unbeaten 70 and picked up six wickets in the second four-day game.
The first Test begins on 15 August in Galle and the second and final game of the series set to start on 23 August at the SSC in Colombo.
India Test squad: Shubman Gill (captain), KL Rahul (vice-captain), Yashasvi Jaiswal, Rishabh Pant (wk), Sai Sudharsan (subject to fitness clearance), Dhruv Jurel (wk), Ravindra Jadeja, Kuldeep Yadav, Manav Suthar, Jasprit Bumrah (subject to fitness clearance), Mohammed Siraj, Prasidh Krishna, Gurnoor Brar, Devdutt Padikkal, Saransh Jain
India interim head coach VVS Laxman has praised Vaibhav Sooryavanshi for his maturity after the teenager scooped the player-of-the-series award in the three-match T20I series in Zimbabwe.
It was the toughest match of the three-game series but India emerged from the third and final T20I against Zimbabwe with a 35-run victory to win the series 3-0.
India batter Ishan Kishan revealed he ignored suggestions from the team that they should be aiming for a score of 250 after they had lost early wickets in the second T20I against Zimbabwe.
India wrapped up a T20I series victory in Zimbabwe with a game to spare after thrashing the hosts by 90 runs in the second match at the Harare Sports Club on Saturday.
