B737 Captain reported the flight attendants reported a smell of smoke and electrical burning during engine start. Flight returned to gate for maintenance action.

Narrative

After we started the number 2 engine the aft flight attendants called and reported a smell of smoke and electrical burning. The flight attendant advised that the smell started in the aft section of the aircraft and worked its way up the cabin. We notified operations that we needed to return to the gate due to the smell of smoke and electrical burning; and decided it would be best if we had as a precaution crash fire rescue come out in the event that there was an issue. We made a CTAF announcement to CFR (Crash Fire Rescue) and heard nothing in response and after seeing that no one was there to park us at gate X we decided in the interest of safety to let operations know that we were most likely going to request priority handling with ATC and we need to be parked ASAP. It took a few minutes for us to be parked which was alarming; giving the fact we told them there was the smell of smoke and electrical burning. I'm not 100% clear if we did request priority; but I know after parking the jet and completing all required checklists it was reported to us that one of the fire men stated that he could smell something from the jet bridge and once he got on the aircraft. We wrote a maintenance logbook entry and filled out the the smoke checklist and it was decided that we would do an engine run which we had authorization from the Chief Pilot and Maintenance Control. We completed the engine run with maintenance on board and it was reported to us that the maintenance technician in the aft of the aircraft Once we started the engine could initially smell something; but that quickly dissipated. After completing the engine run I had a conversation with our flight attendants; and they conveyed that they were uncomfortable flying the aircraft. I then decided to consult the Chief Pilot with the issue and had another conversation with the flight attendants regarding the condition of the aircraft; and it was conveyed to me that their issue was now related to rest; we initially had a passenger issue and now a maintenance issue. There was one mechanic who in my opinion seemed to be very pushy; somewhat disrespectful and abrasive. I felt that we were being pressured into flying in this aircraft even after we had told maintenance that around row three and four there was a smell of electrical burning again; but it was momentarily. Again contacted Maintenance Control and explained to them that around row three and four the smell was coming and going periodically and I was told that the plane was gonna be down and told the next maintenance crew came around XA AM. One of the mechanics was talking in a tone that I didn't appreciate or feel was professional towards the flight attendance stating that there's nothing wrong with the airplane and that he didn't smell anything and it got to the point to where he was becoming combative and somewhat argumentative and making the crew uncomfortable. We were then told that maintenance had gone home for the night and then when I went for the jet bridge; the gate agent told me that only one of the mechanics went home. The gate agents left the jet bridge unattended and it was reported to me that twice the gate agents made PAs to the passenger stating that the flight crew didn't want to fly the airplane and their handheld mics were turned up so loud that people could hear over their speaker that someone was yelling stating do not tell the passengers anything. This was all conveyed to me by a non-revving supervisor. I felt this whole situation could have been handled better; when you have the smell of smoke and electrical burning this should be taken seriously and I felt they were pressuring us into flying the jet. Maintenance should never be rude or abrasive and they should never argue with anyone on the crew. The one mechanic in my opinion needs to receive some training on his professionalism and how to deal with co workers. I've never experienced this in my time at this company. Operations should have told the groundcrew what our issue was but they didn't which was told to my first officer by a ramp agent. The gate should have never made a factitious announcement stating the crew didn't want to operate the jet; that was never said and that placed us in an uncomfortable position as a passenger engaged a few flight attendants outside having some very harsh words until they were told what had actually happened. In this day and age with more and more passengers assaulting airline employees the gate should have known better than to make false PA's. It was my understanding that a supervisor witnessed this and spoke to the gate about this. As we left the jet I explained to the agent that the jet was not operating until at least XAam which is what I thought Maintenance Control stated. I apologized and left with my crew.