The clubs have reached terms on a transfer worth up to £120m (€140m). Diomande will sign a seven-year deal with the Spanish giants until the summer of 2033.
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Real Madrid had a previous £85.4m (€100m) bid rejected for the 19-year-old, with Leipzig wanting significantly more to let him go - which they have now secured.
Jose Mourinho's side have been active in this window, having already signed Bernardo Silva, Ibrahima Konate, Denzel Dumfries and Marc Cucurella.
It comes after Paris Saint-Germain ruled themselves out of signing Diomande. The Ivory Coast international had previously indicated a preference to join the Ligue 1 side earlier this summer and agreed a contract with them until 2031.
However, in a statement given to Sky in Germany, the club formally withdrew their interest, saying the "asking transfer fee and salary demands were completely disproportionate and distortive".
Liverpool also expressed an interest in Diomande, but were not willing to pay more than £85m. Premier League clubs Arsenal and Manchester City were also keen on the Ivorian.
Diomande interview: Leipzig & trials at Palace
"Everything went fast," says Yan Diomande. Indeed, it is extraordinary now to think that this time last year Diomande's entire senior career amounted to half a dozen starts for Leganes at the back end of last season as they were relegated from LaLiga.
He scored in two of those six games, against Espanyol and Valladolid. Even his team did not find the net in the other four. But the teenager did enough to persuade RB Leipzig to part with €20m to bring him to the Bundesliga. Once there, he has been a revelation.
Thrilling to watch and impossible to defend, Diomande is lightning, full of unpredictable enterprise. He has the things that cannot be coached and by listening to the things that can he is getting better. The very biggest clubs want him. The rest cannot afford him.
Read Adam Bate's full interview with Diomande here...
