Dr Rizwan Mustafa, who heads up police training at Staffordshire University, shared content promoting the destruction of Israel and described Jews as “filth”, according to an official report on the Government's anti-terrorism strategy.
West Midlands Police and Crime Commissioner (PCC) Simon Foster launched an investigation in February after saying he was "deeply concerned" about the matter.
He said he had instructed West Midlands Police to report to him "as a matter of urgency", but more than a month later the probe is remains ongoing.
Dr Mustafa holds a senior role at Staffordshire University, where he runs courses and lectures police recruits from across the region.
Nicola Richards, Conservative MP for West Bromwich East and chair of the APPG on combatting anti-Semitism, said she had written to Mr Foster urging him to ensure the posts would be treated as a hate crime.
She also called for the probe to look at whether Dr Mustafa shared extremist views or content with other police officers during his time with the force, and for a full review of what information he had access to, given his security clearance.
Ms Richards said: "It was deeply troubling to learn of the alleged views of Dr Rizwan Mustafa, which are reported to include anti-Semitic social media posts and conspiracy theories around terrorism.
"This is extremely concerning given West Midlands Police relied on advice from Dr Mustafa on extremism, not least that he is a former police officer and now teaches at a university.
"I was reassured when the PCC appeared to be quick off the mark to launch an investigation, but 34 days on we still have no answers.
"I am disappointed with the lack of urgency from the PPC, as well as the refusal so far to confirm to me that he will also be investigated for alleged hate crime.
"The public and the Jewish community need to know that this matter is being taken seriously. It’s time that we saw some action, not just words from the PCC."
The posts were highlighted in the Shawcross review into the Government's anti-terrorism strategy Prevent, which was published by the Home Office in February.
A spokesman for Mr Foster said: "The investigation is still ongoing. We cannot say any more than that at this moment."
In recent weeks Mr Foster has been involved in a bid to become Labour's candidate for West Midlands Mayor, which collapsed after he was rejected by party members.
Staffordshire University adopted the International Holocaust Remembrance Alliance (IHRA) definition of anti-Semitism in January 2021.
At the time the university said it was "a crucial step in combatting prejudice and makes clear that anti-Semitism will not be tolerated at our institution."