Your villain should be as important as your hero. Their purpose is to be a force that forces the hero to change or commit to themselves to become beyond what they were at the start of the story.
If a villain is to be redeemed, then it must be part of what the hero is striving for, and it shouldn't happen organically outside of the result of the hero's arc.
Its possible for the villain to have this arc if the hero also mirrors it--
For example:
Assume Luke Skywalker is the hero, Darth Vader is the villain.
But at different points, there are other supporting protagonists and supporting antagonists that allow the hero and their "party. Han vs. Jabba, Vader vs. Emperor, Rebellion vs. Empire. In the scope of the Luke vs. Vader, however, what starts of as Luke's journey to defeat Vader, the terms of what "defeat" mean change over Luke's arc. Vader's arc doesn't change until Luke "wins" at which point he is redeemed.
This can only be accomplished by shifting the meaning of Defeat = Kill into Defeat = Turn to the Light side.
Now for a poor example:
Assume Luke Skywalker is the hero, Darth Vader is the villain.
Luke has the goal to kill Vader, they battle, Luke loses his hand, etc.
Rematch, Luke tries to kill Vader--he does not think he can be turned back to light--at the very last moment when Luke is about to die at the hands of the Emperor, Vader is redeemed.
This isn't a satisfying arc for Luke because he doesn't accomplish his goal, and Vader's redemption isn't satisfying because it isn't seen through the lens of the Hero's struggle.
In conclusion
A villain cannot be redeemed unless it is the natural conclusion, or the "reward" of the hero's journey.
If the villain is redeemed but it is not the consequence of the hero's journey, then SURPRISE--your villain is actually the protagonist and the "hero" you've been writing about all along is actually just a supporting character.
a villain always sees themselves as a good guy doing necessary evil, usually for the sake of themselves, or the greater good. Rarely will a villain be evil for the sake of being evil.