A Sega exec tried to push Microsoft to make the Xbox backwards compatible with the Dreamcast even [1], which Microsoft refused mostly because they didn't want to support online infrastructure for titles that used it. The original Xbox seemed to have been considered to be the favored successor to the Dreamcast by Sega at least at one point.
Then again, Sega's main initial portwork was focused on the GameCube. The Dreamcast games Sonic Adventure 2, Ikaruga, Crazy Taxi, Phantasy Star Online and Skies of Arcadia were ported to the GameCube and not to the Xbox. They also brought originally-for-Dreamcast titles Super Monkey Ball, Beach Spikers and Billy Hatcher and the Giant Egg to the GameCube exclusively.
Then again, Sega's main initial portwork was focused on the GameCube. The Dreamcast games Sonic Adventure 2, Ikaruga, Crazy Taxi, Phantasy Star Online and Skies of Arcadia were ported to the GameCube and not to the Xbox. They also brought originally-for-Dreamcast titles Super Monkey Ball, Beach Spikers and Billy Hatcher and the Giant Egg to the GameCube exclusively.
[1]: https://kotaku.com/how-xbox-could-have-helped-the-dreamcast-...