Air carrier Captain reported that during pushback at JFK another aircraft under tow came very close to causing a collision.

Narrative

While pushing back from the gate in JFK while still connected to the tug; an [air carrier] wing that was under maintenance toe via a super tug; came very close to the tug that was towing us.Cause: Before pushing the crew came up to the cockpit and briefed me that their headset was broken and that they could hear me; but I would not be able to hear them; and that we would use partial hand signals and partial verbal commands; which I agreed to since we briefed at a time face-to-face. While pushing back from the gate; the tug crew was running behind and then when they pushed us; we were cleared to push a beam the gate. at the same time or after we pushed an [air carrier aircraft] that was under a super tug being pulled by their Maintenance was cleared to reposition by Ramp Control. We were pushed back onto the line and then we were starting to be pulled forward so that we would be a beam the gate as instructed by Ramp Control. The [air carrier aircraft] that was under tug I don't think was paying close enough attention to where their wing and I did not see any wing walkers was because I saw them getting closer as they were pulling it to a gate which was to our left. I saw the gate that I thought that they may be heading to and as they closed closer; I told my push crew to stop pulling us forward. I had to yell stop to them multiple times before we came to a stop. and they finally did come to a stop just prior to me; stepping on the brakes and setting the parking brake to stop us being pulled forward; but I did not want to break the shear pin unnecessarily. The [air carrier aircraft] left wing came very close to the tug that was pulling us. Maybe if I had to guess about 6-8 feet is what it looked like from the cockpit; I believe that if I would not of made the call to stop as I did; the wing would've been even closer or perhaps even over top of the tug that was pulling us forward or worse. I made a call to ramp after we had come to a stop after wards and ramp did not say anything back; but the [air carrier aircraft] crew made a ramp call on the frequency stating they had been cleared to reposition… since there was no collision we just started up and continued on normally.Suggestions: I'm not sure what could potentially avoid this. It seems like we were given a clearance at the same time and almost potentially occupied the same spot on the ramp other than having the [air carrier aircraft] maybe have some more observers watch their wing tips while they move or just being more careful while the repositioning; I'm not sure what else can be done since that ramp area is so incredibly congested and busy 24 seven.