The Mobile municipal court website lists a third-degree harassing communications charge against Cousins, but no other details were disclosed.
USA Today, citing a defense attorney not involved in the case, reported the likely outcome of the warrant would be that Cousins' representative would arrange a time for the player to turn himself in to police.
The NBA and the Lakers said on Tuesday they were investigating a report that Cousins had threatened to shoot his ex-girlfriend in a leaked audiotape.
The recording was published by the TMZ.com website.
In the recording, a man identified as Cousins by his ex-girlfriend Christy West, threatens her after she refused to allow their son to attend his wedding in Atlanta.
"I'm gonna ask you this one more time before I take it to another level," the man identified as Cousins says. "Can I have my son here, please?"
When West refuses, the man warns her he will "make sure I put a bullet in your head."
West, who lives in Mobile, gave a statement to police there and told USA Today that a warrant had been signed.
"We are aware of the allegation involving DeMarcus Cousins and, of course, take this claim seriously," the Lakers said on Tuesday.
"We are in the process of gathering information and will reserve further comment at this time."
Cousins, 29, signed a one-year contract with the Lakers on July 6 but tore knee ligaments during pre-season earlier this month, meaning he faces an indefinite period on the sidelines.