A spokesperson for the force said: “We have received an allegation relating a social media post and are assessing the information.”
In his statement on X, external, Callaghan said he created a video about the Billericay Conservative Party.
Regarding the antisemitic content, he said: “I was completely unaware of this at the time I posted the video and, to my enormous regret, did not check the video adequately before posting.
“As soon as I was made aware of the antisemitism within the lyrics, I immediately deleted the post.
“Antisemitism has no place in our society and I am deeply sorry for unknowingly sharing something that carried antisemitic content.
“I would never have posted it had I known, and accept that I should have been more careful in my use of AI.
“I will learn from this mistake and be far more cautious in the future.”
His video included the caption: “The Billericay Tories will run Basildon Council again in 35 days time if Reform UK win enough seats to put them back in charge.”
There are elections to Basildon Council on 7 May.
Jackson came under fire for the lyrics in the original version of They Don’t Care About Us in 1995.
At the time, the singer apologised, stating he had intended to demonstrate the hatefulness of racism, before re-recording the song and changing the lyrics.