Air carrier flight crew reported a ground conflict with another aircraft during pushback from gate.

Narrative

During pushback from Gate XX we experienced company 737 turning into us from Gate XY and transitioning to Taxiway 1 en route to Spot XX. It appeared to me that the aircraft could be a conflict and I asked the Tug Driver to confirm our clearance with this outbound aircraft. At the same time the Wing Walker on the left side walked toward the Tug Driver and may have jestured; but I did not observe a stop signal. As I was moving to be able to apply brakes the Tug Driver stopped the push. The outbound aircraft maintained an angle toward us as he proceeded quickly to intercept Taxiway 1 eastbound. I was less than comfortable; and the Tug Driver commented that 'that was close!' After company had passed we continued normal push and remainder of flight was normal. We were just starting a trip; fatigue no factor; were SOP; and normal lighting used during push. I did not hear Ramp Clearance to company aircraft but FO (First Officer) does not remember Ramp mentioning our push nor did they direct direction of turn for company. There were no aircraft within several gates of the 2 aircraft involved.