There were strong rumours Reid was considering quitting over the weekend despite the 2-1 win at home to Blackburn.
Reid was furious after McKenzie conducted a public poll, asking supporters if he should be sacked or stay on after three defeats in a row.
Reid said in The Sun: "I've had a chat with the chairman and I'm happy to carry on. We have reached an understanding and I have to say I never thought about leaving. Quitting is not in my nature. I have impressed upon the chairman that, whenever club business like this is done in future, it should be carried out in private.
"Following our talks I am happy to remain as manager and I honestly feel our 2-1 win over Blackburn last Saturday can be the platform for us to kick-start our season."
Reid's failure to attend a post-match press conference following Saturday's win added fuel to the fire about him leaving. Now he has received assurances that there will be no repeat of last week's events and he has the board's unconditional backing.
But he added: "I just felt if I had gone in after all that had happened over the previous 72 hours, the questions would all have been about me.
"That would have taken away from a terrific performance by my team.
"I was delighted with the way they played against Blackburn. They were fantastic and so were the fans."
The plc vice-chairman Allan Leighton is understood to be less than impressed with McKenzie's actions last week and, to make matters worse, Leeds' final year figures are about to be announced and are expected to be similar to last year's pre-tax losses of almost £34m.