The Barcelona forward made the claims in an interview with Jorge Valdano for Movistar's Universo Valdano, which has now aired in full after several days of clips.
The 19-year-old, who won the World Cup with Spain this summer, is among the 30 nominees for the 2026 Ballon d'Or, including Real Madrid's Mbappe, with the winner due to be announced in London on 26 October.
Asked who should take the award, Yamal said:
"I would give it to the best in the world. Who they're going to give it to is a very long conversation. I sincerely believe that, as the best in the world, there are two of us. And this year I would deserve it for what I've won. Mbappé and I are the best in the world. Everyone who watches the games knows it.
"We have a direct duel and I've beaten him in everything. He has scored a lot of goals, but that's why he's won the Pichichi."
He also argued that the criteria have drifted since the days of Lionel Messi and Cristiano Ronaldo.
"This award is being valued in a different way. When Messi and Cristiano were around, everyone knew they were the best, so there was no discussion. The problem is that people are valuing things that aren't part of the Ballon d'Or, like when they say a PSG player has to win it because he has won the Champions League."
He went further on how the winner should be chosen.
"That award has to go to the best player in the world, full stop. Not to a player who has scored 80 goals, because when you score that many they give you the Pichichi. They are different things that people aren't understanding.
"The best player in the world is the one you enjoy watching when you're watching the game, a player different from everyone else. When this award goes back to focusing on what I've said, that is when it will have the value it has had historically."
On whether Spain's World Cup triumph should count towards the prize, he was equally direct.
"But I think the coincidence is that I am in the team that has won the World Cup. If England had won it, I would still be saying that Mbappé and I are the best."
For the best player of the tournament itself, though, Yamal looked elsewhere. He named Messi, whose Argentina lost the final to Spain.
"Even though everyone knows he is the best in history, nobody expected such a legendary performance. For me Argentina has good players, but Leo is superior. For me he was the best player of the World Cup."