L.A. Clippers owner Donald Sterling told his GF he does NOT want her bringing black people to his games ... including Magic Johnson ... and it's ALL on tape.

TMZ Sports has obtained audio of Sterling making the racist declaration during a heated argument on April 9th with V. Stiviano ... after she posted a photo on Instagram posing with Magic.

Sterling rails on Stiviano -- who ironically is black and Mexican -- for putting herself out in public with a black person (she has since taken the pic down). But it doesn't end there. You have to listen to the audio to fully grasp the magnitude of Sterling's racist worldview. Among the comments:

-- "It bothers me a lot that you want to broadcast that you’re associating with black people. Do you have to?" (3:30)

-- "You can sleep with [black people]. You can bring them in, you can do whatever you want. The little I ask you is not to promote it on that ... and not to bring them to my games." (5:15)

-- "I’m just saying, in your lousy f******* Instagrams, you don’t have to have yourself with, walking with black people." (7:45)

-- "...Don't put him [Magic] on an Instagram for the world to have to see so they have to call me. And don't bring him to my games." (9:13)

Sterling has a documented history of allegedly racist behavior -- he's been sued twice by the federal government for allegedly refusing to rent apartments to Blacks and Latinos.

He was also sued by former Clippers exec Elgin Baylor for racial discrimination -- though a jury was ultimately not convinced and shot down Baylor's case.

Sterling has been separated from his wife Shelly for years. She remains a key player in running the team and sources tell us she's "mortified" by Sterling's comments.