Air carrier Captain reported a ground conflict with the pushback tug shortly after initiating taxi. The Captain reported he failed to ensure the tug was clear before calling for taxi.

Narrative

We received pushback clearance off gate X to push long and stop abeam gate Y. Pushback commenced and tug crew cleared us to start engines. Engine start was completed before pushback was complete. Once pushback was complete; brakes were set and the tug crew was cleared to disconnect. Tug crew announced they were departing to the right; FO side. While completing the after start checklist checklist I announced the tug crew was departing to the FO (First Officer) side. With the after start checklist complete we delayed calling Ramp Tower for taxi due to a 767 in front of us that had yet to begin taxi. This was the same aircraft that required our long pushback. Once the 767 started rolling we asked and received taxi clearance. I announced clear left; the FO responded clear right and we began to taxi. After approximately 15-20 ft. and 2-3 GS (Ground Speed); we made physical contact with the pushback tug. I immediately stopped the aircraft; then set the parking brake. I had the immediate realization that we made contact with the tug. Tug crew announced that the aircraft nose tires made contact with the tug. Tug crew announced they were OK.A breakdown in HF/CRM (Human Factors/Crew Resource Management). I failed to ensure the tug crew was clear of the aircraft before calling for taxi. I perceived the FOs 'clear right' as the tug crew was clear. In retrospect; I now know I never heard the FO announce specifically the tug crew was clear nor did I observe the tug crew clearing off to the right. In my mental checklist; I checked off that the tug crew was moving away from the aircraft and the FOs clear right call confirmed the tug crew was indeed clear. I accept full responsibility the the dangerous situation I created. I own the parking brake. I further apologize for all the additional work and time others will be required to provide to correct my mistake.Suggestion: Strict adherence to the [company procedures]. Do not taxi or call for taxi clearance until maintenance is verified to be clear of aircraft.