Mbappe, who starred for Paris Saint-Germain against Barcelona in the Champions League on Tuesday, spoke about meeting one of his idols in the game when he was a teenager.
Drogba was one of the best strikers in the Premier League during his peak in the mid-2000s.
"When I was 11 years old, I got to go to London for a few days to train with Chelsea's youth team," wrote Mbappe in a column for The Players' Tribune.
"I was so excited and shocked that I didn't even want to tell my friends from the neighborhood where I was going.
"When I came back home, my friends saw me and said, 'Kylian, where were you last week?'
"I said, 'I was in London with Chelsea.' They said, 'Pfff, no it's impossible.'
"I said, 'No, I swear, I even met Drogba.'
"They said, 'Pfff, you're lying. Drogba doesn't meet kids from Bondy. It is not possible!'"
"I didn't have a phone back then, so I asked my dad to give me his, and I showed them the pictures we took," Mbappe added.
"That's when they finally believed me. But the important thing was that they were not jealous. Not at all.
"They were just amazed I will never forget what they said to me. I can still picture it, because we were actually in the changing room at AS Bondy [Mbappe's youth team], getting dressed for a match.
"They said, 'Kylian, can you take us there with you?'
"It was like I had gone to another planet. I said, 'But the camp is over now. I'm sorry.'
"They were looking down at the phone, laughing and shaking their heads. They said, 'Wow. It's like we are living this moment with you, Kylian.'
"That's how much it means to us. Going to these places, it's like visiting another planet."