Im pretty certain they've done this, and he has already told them "no." Plus, let's just think about this for a second for anyone who may feel he would take the job. He retired from a dynasty that he created, and was still getting us championships and getting into the playoffs. He walked away from that, and that's due to the current landscape of college football. And it's gotten worse imo, why would he want to come back into something that he originally hated so much, it made him retire? And not to mention, for LSU? Hell no. Especially not now, they have completely fucked themselves over with how they have handled this whole coaching change. I've seen/heard LSU fans bring up "his love for LSU", and how he regretted leaving from there. That's not the case, he regretted leaving college football, he missed developing players on and off the field and having the gratification of doing so. That why he ultimately came to Alabama, to rectify that regret. And boy did he ever.
And just a few more points. Going back to the LSU fans and their claims about him loving LSU so much. Saban and his family spent 17 years in Tuscaloosa, they are fully engrained into that place. WAY more than LSU. Not to mention the statue, the field being named after him, the new facility that they're building for the youth that's in his name. Again, fully engrained. Plus, he's 74 years old. Has infinite wealth, done everything you could possibly do in this sport as a head coach.