The chief told The Times-Picayune that he was disturbed by the post.

“I will tell you this: This will not go unchecked,” he said. “I’m not going to take this lightly and this will be dealt with on our end. It’s not something we want someone that’s affiliated with our department to make these types of statements. That’s not going to happen.”

The chief said that the post appeared to violate the department’s social media policy but that he did not think it rose to the level of being a threat.

Ms. Ocasio-Cortez is part of a quartet of first-year Democrats in the House known as “the squad” who have received death threats and were told by President Trump to “go back where you came from” in a racially charged post on Twitter.

Belinda Constant, the mayor of Gretna, said on Sunday that she had been out of the state and was not immediately prepared to comment. “I will be looking into the situation,” Ms. Constant, a Democrat, said.

Eva Malecki, a spokeswoman for the United States Capitol Police, declined to say whether the agency was investigating.

“The mission of the United States Capitol Police is to protect and serve Congress,” she said. “However, we do not discuss how we carry out our protective responsibilities.”

The four congresswomen were also the subject of a recent Facebook post by the Republican County Chairmen’s Association of Illinois, which depicted them in a mock movie poster that referred to them as the “jihad squad.”