In 1487, officials in Milan asked architects to submit design ideas for a tiburio, lantern tower, for the top of their under-construction cathedral. Leornardo collaborated with architects Donato Bramante and Francesco di Giorgio on the design. He worked to design a double-shell dome, very similar to the one in Florence's cathedral. However, Leonardo favored a more traditional style, so he did not try to make his dome match the gothic design of the rest of the Milan cathedral. As a result, his designs were rejected and he eventually withdrew his application. The winning designs came from Francesco di Giorgio, and architect from Siena, who Leonardo worked with on other projects later on. 

Source: Isaacson, Walter. “Vitruvian Man”. Leonardo Da Vinci. Simon & Schuster, 2017

Photo Source: http://www.leonardo-da-vinci.net/architecture/