Air carrier pilot reported the tug came in contact with the aircraft during push back and the aircraft was taken out of service.

Narrative

We were operating flight from ZZZ to ZZZ1. We were departing from gate X at terminal X. Everything was normal up until it was time to push back from the gate. After receiving the ok to push" and we completed all checklists we turned the red beacon on. The ramp crew got our attention over the headset and they did their normal pre push exchange of information but added that they had someone pushing us that was training and we had to let ramp know. First Officer called for push and advised ramp that we had someone training who would be doing the push. Ramp gave us a clearance to give way to the aircraft behind us and then push to spot X. At that point we coordinated the release of the parking brake and gave the instructions to the push crew. They read everything back and confirmed the instructions. After they initiated the push and got away from the gate it seemed like they started to turn us so we were lined up on the inner line of the ramp. When this happened I noticed the tug was very far off the left of the nose which was unusual from what I am use to seeing and saw that the person on the headset looked to be the trainer. Then from our perspective the tug disappeared under the nose of the aircraft. It was completely at a 90 angle from the nose of the aircraft moving from left to right. From my experience driving the tug in that direction made sense but they seemed very close and under the nose compared to what I have witnessed as normal. At that point they stopped moving the aircraft. The trainer came on and stated they needed to bring us back to the gate. I had asked for some more information of why that was needed and he stated that the tug touched the aircraft and because he saw it they needed to have maintenance come look at it. We coordinated with ramp and they pulled us back into the gate. The jet bridge was reconnected and tech ops looked everything over. They stated it was outside the allowed limits and they were taking the aircraft out of service. This lead to us deplaning the customers and doing a tail swap."