The 39-year-old singer has revealed she was initially keen to give the record - which has become one the signature tracks of her career - to Christina, but she was eventually talked out of it by her A&R.
She told Stereogum: "She reached out and was like, 'Would you write something for me?' And then it came through so many emails. What a voice! I know I could write something fire for her.
"So time passed and I hadn't written the song yet. And [the label] was like, 'Do you have anything for Christina?' I remember I had just recently written 'If I Ain't Got You'."
Alicia discusses the idea of sending the 2004 hit to Christina, but the idea was quickly shot down.
The A&R said: "Are you f****** crazy? We're not giving her that song. Are you out of your mind?"
Alicia worked with Christina, 39, on the pop star's 2002 album 'Stripped', and she relished the experience.
The chart-topping singer - who released her debut album, 'Songs in A Minor', back in 2001 - shared: "It was such a cool vibe for us to be able to work together during those times.
"But I will never forget that I would have given away 'If I Ain't Got You'. Isn't that crazy? So that's a good story."
Meanwhile, Alicia also admitted to being inspired by Erykah Badu, revealing she loved her sense of "style".
The award-winning star said: "I never felt more jealous in my life because I was just a shorty trying to figure out how I can craft my music. I couldn't figure out what it was supposed to sound like. And she just seared through in this very specific manner.
"She was so clear about who she was. You could see it in her videos and her style."