3:37 PM PT -- The NBA tells TMZ Sports it is aware of the report and working on getting more information on the alleged incident.
2:43 PM PT -- The judge who signed off on the protective order has ordered Rondo to temporarily stay at least 500 feet away from Ashley and their kids.
Ashley has also been awarded temporary custody of the two children.
Rajon has also been ordered to temporarily turn over his firearms to the court.
FWIW, the court's granting of the Emergency Protective Order isn't an indication of the ferocity of the claims within it. It's simply a temporary order until the parties can get before a judge and argue the merits of a protective order.
Rajon Rondo allegedly pulled a gun on the mother of his children and threatened to kill her during a heated altercation ... and the woman claims their kids witnessed the whole thing.
TMZ Sports has learned 36-year-old Ashley Bachelor, Rondo's former partner and mother of his two kids, filed for an emergency protective order in Louisville last week, requesting protection for her, her son and daughter ... and the story she tells about the alleged incident is scary.
Bachelor says Rajon -- a 2x NBA champ and 4x All-Star -- was playing video games with his son on May 11 ... when Ashley asked the boy to separate laundry. Bachelor says Rajon became enraged and ripped the game console out of the wall in front of the kid.
Ashley says Rajon then continued the destructive behavior, smashing everything from a teacup to outdoor lights and trash cans.
Bachelor says the kids were upset after witnessing their dad losing it, so she tried to "deescalate" the situation. But, she says Rajon responded by making a death threat, allegedly saying, "You're dead."
Rondo temporarily left the home, according to Ashley, before returning about 15 minutes later, when he allegedly approached a back door and began banging on the window with his gun.
Bachelor says she didn't know what was going on, so she approached the door from the inside to take a look. She allegedly saw Rondo with the weapon and asked him to stop because he was scaring her. She says Rondo didn't listen, and instead yelled, "Go get my f**king son!"
Ashley says she feared if she didn't comply with the alleged gun-wielding hoops star something bad may happen, so she brought their son, Pierre, downstairs, but kept the backdoor closed.
Soon after, Ashley says Pierre opened the door and Rondo "yanked him outside."
Bachelor says Rondo, who was still allegedly holding his gun, was screaming at Pierre, asking why the boy was scared of his dad.
Ashley, who claims her son was "visibly upset," says she didn't intervene because she was scared Rajon would shoot her.
Bachelor says Rajon eventually demanded their daughter also come outside ... where she claims he yelled at both kids for being afraid of him.
Thankfully, Ashley says Rajon's mom and dad both arrived at the home ... and although they didn't have much success in calming down Rajon, she claims he eventually left the scene.
But, now Bachelor says she's scared for the safety of her and her kids, writing the following in the emergency order of protection on May 13.
"I am extremely fearful for my safety and for the safety of my children. Rajon has a history of volatile, erratic, explosive behavior. He is verbally, emotionally, and financially abusive. He physically hits our son and calls him names like 'p***y' and accuses him of acting like a 'bitch.'"
"Rajon verbally assaults our daughter. He calls her names like 'thot, bitch, and d**khead.' Rajon has made several threats on my life, saying at various times he will shoot me or shoot up my car."
Ashley Bachelor was granted protection for herself and her two children by a local judge.
We reached out to Rondo's reps, but, so far, no word back.
Based on our digging, Rondo has not been named a suspect in any crime by police, and has not been arrested.