The Dutch giants have endured a horror start to the season, with the club languishing in 13th spot in the Eredivisie table after four games.
Steijn, who was only appointed at the Johan Cruyff Arena in the summer, has won just one league match.
And the 49-year-old knows he will be under pressure should his side fail to claim all three points against rivals Feyenoord.
According to De Telegraaf, ex-Arsenal transfer chief Mislintat is already gunning for his dismissal.
The Dutch news outfit has claimed that 'Diamond Eye' stormed into the Ajax trainer's room at the club's training complex to discuss the possible axing of Steijn.
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He is alleged to have made the request on Friday - two days before The Classic.
His demands are said to have been met with anger amongst the staff members present, as well as by surprise.
It was only on Wednesday that Ajax announced that Mislintat was being investigated following the signing of Borna Sosa on deadline day.
The Croatian defender was signed through an agency that has a stake in Mislintat's private firm.
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An Ajax statement read: "Before the appointment of director of football affairs Sven Mislintat, AFC Ajax NV had been informed by him about his shareholding in Matchmetrics GmbH.
"Contractual agreements were made about this between the club and Mislintat at the time.
"The club management was asked about a shareholding that AKA Global GmbH, a consultancy firm for professional athletes, allegedly acquired in Matchmetrics.
"Ajax was not aware of this. Since a player signed by Ajax was assisted by AKA Global this summer, the club is looking into this further and is being assisted by external advisers.
"An independent external investigation will also be conducted by a forensic accountant.
"Sven Mislintat has stated that it will provide full cooperation, including sharing all relevant documentation."
De Telegraaf also add that the majority of Ajax's members were of the belief that not only should he be suspended pending the investigation, but that he would be working from the shadows.
Nevertheless, his request for Steijn to be sacked will do little for the confidence of the manager ahead of a huge match.