The educator is said to have been photographed holding the mug in the playground at Colchester Royal Grammar School (CRGS).
The Essex school said that “robust action” would be taken as necessary.
“We have been notified an image has been shared online of an individual appearing to use a cup that has an offensive image on it,” a school spokesperson said.
“At this time we are looking into the matter. While we cannot comment on individuals, this will be reviewed in line with our policies and procedures and the appropriate robust action taken as necessary.”
It was reported that the image on the cup was from a cartoon series called “Jesus And Mo”, which was created by a British author using the pseudonym Mohammed Jones, and has been the subject of numerous previous controversies in the UK.
Many Muslims consider depictions of Muhammad to be highly offensive.
Outraged protests were sparked at Batley Grammar School in Yorkshire last year after a teacher showed pupils a cartoon by the French magazine Charlie Hebdo in a religious studies lesson about freedom of expression.
The teacher was suspended, but was offered his job back after an independent external investigation concluded that he had not intended to cause offence by showing the image.
However, the unnamed teacher, who had gone into hiding, was reported at the time to be fearful of returning to the school.
While CRGS was ranked the UK’s ninth-best selective school by the Sunday Times last year, Ofsted inspectors rated it “inadequate” – after finding during an unannounced inspection that children had been “subject to sexist or racist comments”, leading “a significant number” to feel unsafe.
Ofsted’s report in July warned that attitudes, behaviour and personal development at the grammar school were inadequate, with school leaders found to have failed to address a “pervading culture in the school which does not promote equality and respect”.
Following a monitoring visit in February, inspectors said that leaders were taking “effective action”.
Headmaster John Russell said in response at the time that the February inspection showed “the actions we are taking are effective and having a real and tangible impact”, with inspectors applauding the work undertaken “at every level of the school”.
The school was also accused last April of having a “toxic and ubiquitous rape culture”, in a blog post by former student Scarlett Mansfield.
Mr Russell said at the time that he was “shocked and saddened” by reports of “wholly unacceptable behaviours and attitudes” that some female students experienced during their time at the school.