Small Aircraft Flight Instructor reported that during taxi the wing of a transport category aircraft may have passed over the horizontal stabilizer of the smaller instructional aircraft. Reporter stated that IND Ground Control did not notify the large aircraft of the position of the small aircraft.

Narrative

Approximately XA00; the student [Aircraft X] was taxiing to depart IND while the instructor was monitoring. We were given instructions to taxi to 23L via M; M3; cross 32; N; and to hold short of D. While approaching 32 ground was coordinating with an Aircraft Y plane on D just short of N trying to return to ramp; and Aircraft Z (air carrier) plane on 14 trying to get to point X. Initially; ground control told the Aircraft Y plane to hold short so the Aircraft Z could turn right onto D then left into point 2; but Aircraft Z crew offered to continue down 14 and enter the ramp at point 1 which is essentially where our taxi route was. Ground hesitated then approved this request just about when we were crossing the hold short lines to cross 32. We noticed Aircraft Z began speeding up their taxi and the student noted hesitation whether to continue further onto the runway. Since we had already begun crossing the hold short lines I instructed her to continue but to expedite taxi. At this point; Aircraft Z was still just past D with what I had thought would be plenty of forward visibility to see us crossing directly in front of them. As we pulled onto N I became concerned Aircraft Z may not see us as the amount of thrust he was using and my perceived distance he was behind us indicated to me he was expediting his taxi into the ramp. As we approached our hold short instruction at D I radioed ground to confirm Aircraft Z had us in sight. At this point; I don't even think Aircraft Z was still on ground frequency even though he was still on N which is what I was hoping for to give him notice of our position if he was unaware. Before ground responded Aircraft Z began their turn into their ramp at point 2 off N as we came to a stop holding short of D. I was having a hard time judging how far they were behind us due to lack of rearward visibility; and was close to increasing throttle and continuing on delta without clearance but did not. It is hard to tell; but as Aircraft Z turned it seemed to me and the student their right wing crossed over our horizontal stabilizer or at least very close to it. This is what in hindsight seemed to confirm my suspicions they never saw us. After a short delay as we were holding short ground responded something to the effect of; 'Um; Aircraft X yeah the Aircraft Z is clear of Delta so you can continue taxi Delta to 23L.' This indicated to me he did not really understand what I was communicating and was likely unaware of the hazard that had just occurred.In hindsight; I could have cued up ground a little sooner as to my concerns. I wrongly assumed the Aircraft Z saw us. I also should have simply taken control and taxied onto D without clearance and immediately notified ATC as there was no conflict on D. It is entirely possible the Aircraft Y did see us; and spacing was fine - hard to tell with limitations in rearward visibility; but either way it made both instructor and student extremely uncomfortable. If Aircraft Z did not see us I believe there is some responsibility on their shoulders of seeing traffic as we were off their nose at least 1;500 feet ahead of them when they accepted altered taxi instructions; and I believe they deviated from their taxi routing as they were cleared to point 1 but followed us on N to enter at point 2 creating the hazard. Also; ground control seemed unaware of the possible conflict giving Aircraft Z a taxi route without confirming our position along our route and without confirming Aircraft Z had us in sight. I'm also not sure whether they landed on 14; or if it was coming out of maintenance and they were taxing to ramp. I would think the latter as I think I'd have seen them land if so as a 14 landing would be unusual for the winds and traffic.The instructor should have been more proactive in contacting ground earlier upon possible conflict; and should have deviated per PIC authority from clearance to be 100% safety of flight was not compromised. Aircraft Z should be more aware of potential for not identifying small aircraft on taxi; and deviated from their ATC taxi clearance creating a hazard. Ground control should not accept taxi routing suggestions from pilots without being 100% certain they do not create a hazard; and confirming all prior traffic will not be at risk. Ground should have confirmed Aircraft Z had us in sight if they were going to be closely following our routing. I do think it would be pertinent to notify ATC of this event for them to play back and learn from this as well.