Maintenance technician reported a near miss while tugging an aircraft from the gate to hangar when an aircraft taxied in front of them. The technician applied brakes stopping their aircraft to avoid a collision.

Narrative

I was task with the aircraft movement of Aircraft X from Gate XX to the hangar. we were cleared to push off the Gate XX by South ramp frequency XXX.XXX. South ramp cleared the taxi with the Supertug with Aircraft X to Spot XX. West ramp frequency XXX.XXY cleared us to Spot XY. At Spot XY we contacted ZZZ FAA Tower at frequency XXX.XXZ to request to move Aircraft X to the line Maintenance hangar; was told to hold at spot XY. Aircraft Y taxi to Spot XZ an stop. At approximately at XA:20 FAA ZZZ Ground Control cleared us to taxi with the super tug on [Taxiway] 1 and hold short of RWY X/XX. I started to taxi Aircraft X with the super tug. I saw that a few seconds after that Aircraft Y also started to taxi in front of me. I stop the taxi of Aircraft X by applied my breaks to stop the super tug from hitting Aircraft Y. I believed Aircraft Y was on the Ramp frequency and I was on FAA ZZZ Ground Frequency and could not hear what instructions that each other was given. The FAA ZZZ Ground Control was working both Ground frequency and was busy handling their work load.