Captain Marvel has used a handful of different names in the comics. Originally appearing as Ms. Marvel, Carol Danvers would use the codenames Binary and Warbird as well, before eventually settling on Captain Marvel. However, when she transformed into Binary, she was offered - and declined - a spot on the X-Men.
In the comics, Carol Danvers had a traumatic transformation to becoming Binary. Rogue, acting as a villain before she became a superhero, permanently stole Captain Marvel's abilities and memories when she was Ms. Marvel. While her powers would later be restored, Carol would become Binary after getting experimented on by the Brood, giving her even more cosmic powers. When she first unleashed her new abilities, she impressed the X-Men and got an invitation to join.
Uncanny X-Men #164 by Chris Claremont, Dave Cockrum, Bob Wiacek, Janine Casey, and Joe Rosen features the X-Men and Carol Danvers facing off against the Brood in space. Carol uses her newfound powers as Binary to unleash a powerful energy blast to save the team by turning the warp-drive back on and reactivating the spaceships life support. The X-Men are impressed and while she's helping repair the ship with Colossus, she's invited to join the team.
Mutant or not (and the X-Men have had plenty of non-mutants on the team), Colossus sees Carol's value and makes the offer. While she's flattered, Carol ultimately declines the offer as she's always been fascinated about traveling among the stars - with her new powers, she can continue her journey. If she joined the X-Men, she'd have to reside on Earth, which she fears has no place for her new persona, Binary.
Carol Danvers would continue to work alongside the X-Men and even sparred with them soon after the offer. Carol made a dramatic return to the X-Mansion as Binary just a few issues later, punching Rogue through the ceiling and all the way into space as an act of revenge. After Colossus manages to subdue her, Carol lets Professor X know she's not OK with the woman who destroyed her life becoming a part of the X-Men, and says she'll never be an X-Man. While she might not ever become an X-Men officially, Carol would later become the head of the X-Men spinoff team Alpha Flight and put the bad blood with Rogue behind her. There's no doubt Captain Marvel would have made a welcome addition to the X-Men - they, like the Avengers, really could use her immeasurable power.