DEMI MOORE's two oldest daughters are locked in a heartbreaking battle over their careers and their off-and-on relationship with their mom, a source says.

RUMER, 25, is going full blast with her singing career, angering her 22-year-old sister SCOUT, who's been struggling to crack the music scene with her duo act, Gus + Scout. "Rumer and Scout are at war," disclosed an insider.

This summer Rumer landed a spot at Sayers Club, a Los Angeles celebrity hotspot. "Rumer drew crowds and has generated a lot of buzz," said the insider. "The same can't be said of Scout."

While Rumer is building a loyal following, a recording released last year by Scout and partner Gus Wenner has failed to gain traction. To make matters worse, both young women have had a rocky relationship with Demi, 50, since she landed in rehab early last year following a drug-and-booze collapse.

Adding fuel to the family feud, Demi felt betrayed by Rumer after she secretly stayed in contact with ex-stepdad ASHTON KUTCHER after he and the "Charlie's Angels: Full Throttle" star split, a source says.

Rumer and her mom have since reconciled and now Rumer is urging Scout to forgive dysfunctional Demi, said the insider. But Scout's not listening!

As we reported, Demi failed to show up at Scout's graduation from Brown University this past May. "And Scout still harbors a lot of anger toward Demi over not being a great role model," said the source.

BRUCE WILLIS, the girls' father, has tried to heal the rift between his warring daughters, begging Scout to settle the family feud. "He dragged her along for a few of Rumer's music shows, but Scout wasn't impressed," said the insider.

"She slammed Rumer behind her back. It also isn't sitting well with Scout that Rumer is using Ashton's connections to land gigs."

Said the source: "With everything they've been through with Demi, Rumer and Scout would benefit so much from having each other to lean on."