Air carrier pilot reported ground crew pushed the aircraft back toward another aircraft; resulting in a near collision. After that conflict was resolved; the tug driver then almost pushed the aircraft into a catering truck.

Narrative

After calling for clearance to pushback from Gate XX at ZZZ; Ramp Control instructed us to allow the aircraft pulling into Gate XY to our right to pull in first; and then we were cleared to push back to Spot X. After relaying these instructions to the ground crew; and verifying that the aircraft to our right had parked at Gate XY; our push crew begin pushing us back. At that time; I looked back to my left around the 8 o'clock position and I noticed another Company aircraft had been towed into Spot Y to begin engine start. I mentioned at this time to the FO that I was wondering how they were going to push us back to Spot X with that aircraft in Spot Y; thinking maybe they were going to push us out straight or in a different direction since they (approximately 5 ground ops personnel watching and guiding our pushback) obviously had to see that aircraft there to my 8 o'clock position. Shortly after that; I noticed the tug driver began pushing our tail towards the aircraft in position three; and it was at this time that I asked the tug driver to stop the push because of that aircraft in Spot Y. Simultaneously as I instructed the tow truck driver to stop; the pilots from the aircraft in Spot Y also came over the radio and said we were pushing towards them and that we needed to stop. After stopping the aircraft; I instructed to tow crew to move us back toward the parking gate; and we notified Ramp Control about the conflict; the aircraft in Spot Y said that we had gotten very close to each other on the push back. After this; ramp instructed us to hold position until the aircraft in Spot Y had taxied. While being pulled up to Spot X for engine start; I noticed a catering truck that looked like it could be close to our left wing as we were pulling up. I made 3 different attempts to try to get the tug driver to recognize what conflict I was telling her about and stop until we could ensure there was no conflict. Ultimately; we never ended up stopping; and the driver informed me that the wing walker said we were clear. However; even though we were clear in that situation and nothing happened; I believe the lack of understanding and response from the tug driver and my hesitancy to push on the brakes during the tow would have been too late if the truck actually was a conflict.