Sources with direct knowledge tell TMZ ... since Britney's split with Sam Asghari 3 weeks ago, Britney has been saying she wants to repair the relationship with her dad. She knows Jamie has been ill -- in and out of the hospital for months, and doesn't want the regret of waiting too long.

We've learned Jamie was in the hospital several months ago when doctors believed there was something wrong with his kidneys. Turns out the real issue was his knees ... he had a knee replacement 16 years ago and somehow he developed a terrible infection.

Jamie is going under the knife again Friday, where doctors will mess with the metal in his knees and try to clear the infection. Through it all, he's lost more than 25 pounds and is extremely thin.

We know the only family member she's currently in touch with is her brother Bryan, who has spent time recently at her home. We're told Bryan has filled her in on their dad's condition.

As for Jamie, our sources say he'd love a reconciliation with his daughter. We're told he's never harbored ill will toward her, despite all the things Britney said about him both in and out of court. Jamie learned Tuesday Britney was talking reconciliation, and we're told he was elated.