The France international was caught on camera criticising Neymar after he was substituted and said: "This tramp, he never passes to me."
"Yes, yes, I said it. Now these are things which happen all the time in football. It just needs to be something that doesn't linger," he said. "That's why, right after, given how it blew up, I spoke to him about it.
"We've exchanged a lot of words like that in the past and we'll continue to do so, because we want to win, but there shouldn't be any hard feelings. There is no resentment at all because I respect the player and the man and I admire what he is."
Mbappe also hinted that he is still determined to leave PSG after saying he informed the club of his desire to leave in July amid interest from Real Madrid.
PSG refused to negotiate on a deal for Mbappe in August, despite the 22-year-old having just a year left on his contract. Sources in Madrid previously told ESPN that Madrid made bids of €160m, €180m and finally €200m on Aug. 31 before the transfer window closed.
"I thought that my adventure [with PSG] was over," he added. "I wanted to discover something else. I'd been in the French league six or seven years. I've given what I tried to at Paris and I think I've done it well.
"But everyone is free to draw the conclusions they want, but that's my assessment. Leaving was the logical next step.
"I'm not going to act like a hypocrite and send out a plea, like: 'Oh, I don't know.' This summer my ambition was clear, I wanted to leave and put the club in the best circumstances to bring in my replacement. Right now, my future is not my priority. I've already wasted a lot of energy this summer, it's draining."
When asked about a potential future destination, Mbappe replied: "If I had left this summer, it would have only been for Real [Madrid]."
The striker also said he considered taking a break from international football after missing the key penalty in France's Euro 2020 exit to Switzerland.
Mbappe was criticised for his performances at the tournament after failing to score a goal and also received online racist abuse after his penalty miss in the shootout.
"I have always placed the French national team above everything and I will always put it above everything," he said. "I have never taken a single Euro to play for the French national team and I will always play for my national team for free.
"Above all, I never wanted to be problem. But from the moment where I felt like that I was starting to become a problem and that people felt I was a problem ... The most important thing is the French national team, and if the French national team is happier without me, it is like that.
"I have so much love for the French national team that I abstract from it all. What shocked me, again, was being called a monkey for the penalty. That is what I wanted support around, not because I took my penalty to the left and [Yann] Sommer stopped it."