Tow tug driver reported near collision with a 737 taxiing while towing another 737 aircraft.

Narrative

We were towing a 737 [Aircraft X] to gate X. The tower's instructions were left on [Taxiway] 1 to 2. As we were approaching [Taxiway] 3 called ZZZZ Ramp; the controller instructed 2; 4 and 5 to gate X. We entered the ramp area; as we were joining from 4 onto 5 I turned my head back to my right side to double check clearance between the wing tip and the fence by the service road (As we know it gets too close at this intersection) I have noticed a 737 [Aircraft Y] coming thru on [Taxiway] 6. I immediately stopped because it was getting too close to our wing tip. As soon as I stopped the captain of the 737 slammed on his brakes; stopped and shut off his taxi lights. He was coming so fast that I saw his nose gear diving down as soon as he slammed on his brakes. They almost collided with our plane. I am surprised that he didn't complain at all. I asked my partner to call ZZZZ ramp and say something while we were motionless. I believe she stated that the 737 was instructed to hold short of [Taxiway] 5; also made a comment something like you should have taken 7 and transition to 5 to avoid it'. At the moment there was another oncoming 737 on [Taxiway] 7 being towed towards us. We do not take different routes unless instructed by a controller. Once we confirmed it was safe to move we continued on [Taxiway] 5 towards gate X. As we got close to our gate there was Aircraft Z on [Taxiway] 7 adjacent to gate Y. They called for taxi; the ZZZZ Ramp gave them taxi instructions to taxi without telling them to give us way to go to gate X nor told us to give way to the traffic before turning to our gate. She made two serious mistakes within 2 minutes or so."