The antisemitic posts were posted on Elmo's account in quick succession and called for all Jewish people to be "exterminated".
They also made derogatory comments about Trump and allegations about his connections to Jeffrey Epstein.
The disgraced financier had powerful connections and died in prison in 2019 while awaiting trial for sex trafficking charges.
One of Elmo's posts called on Trump to "RELEASE THE FILES" following an announcement last week that the US Department of Justice and FBI found no evidence of a list that could implicate high-profile associates of Epstein's.
His death is at the centre of a number of conspiracy theories and while he was campaigning to be elected, Trump had promised to publish files about what happened.
The posts on Elmo's account come after X's boss, Linda Yaccarino, stepped down as the platform faced criticism for an increase in antisemitic content.
Billionaire Elon Musk, who owns X, also had to explain last week after its AI chatbot, Grok, praised Adolf Hitler - the Nazi leader responsible for the murder of millions of Jewish people in the Holocaust.