Giggs was working as a coach at United in 2015 when the encounter took place, telling 'Rio Ferdinand Presents: The Debrief' that a fan called Lionel approached him near his hotel to recommend a young Monaco forward.
"I can feel someone, you know, when you feel someone like following you. I get to the hotel and then all of a sudden I can feel someone coming up to me," Giggs said.
"This guy is called Lionel. He comes up to me and says, 'hi, I'm Lionel. I just wanted to just say hello. I'm a big United fan and big fan of yours. But I watch a lot of youth football in France, and you have to look at this player. He's called Kylian Mbappé.
"He's at Monaco and I think he's a player for Manchester United.'"
Giggs said the tip was followed up seriously.
"A few weeks later, he emailed me all the details. So me and Albert, he was another assistant at United at the time, I've told him about this thing," he said. "We go to a few youth games with Monaco and we've seen him and we were just like, 'wow, what is this?' You can see players and they can be special, but he did look exceptional, obviously.
"We told the United scouting team."
The club's scouts watched Mbappé play for Monaco against Paris Saint-Germain, but were left unconvinced.
"They went to watch him. Someone in France went to watch him and he was actually playing PSG. Monaco were playing PSG. So they went to Paris to watch him," Giggs said. "Anyway, I think a few weeks later, I said to this guy, 'how did it go?' He went, 'yeah, they weren't too sure about him, so we just left it.' And that was it."
Mbappé went on to make his name at Monaco before joining Paris Saint-Germain in 2018 for £162 million, then the second-most expensive transfer in football history. He later moved to Real Madrid, where he now plays under José Mourinho, and won the Golden Boot at this summer's World Cup while becoming the tournament's all-time leading scorer.
The same source, Giggs said, later tipped him off about two further French talents who went on to become Premier League regulars: Manchester City's Rayan Cherki and Arsenal's William Saliba.