The Amsterdam club made the announcement on Monday after it was revealed ten Cate held talks with the Blues last week.
Ten Cate will now have to negotiate a contract with the London club. The Ajax statement continued: "Last week, ten Cate informed the club he wanted us to negotiate with Chelsea after he had been approached personally.
Ten Cate was non-committal when asked whether the Sparta Rotterdam clash would be his last in charge with a two-week break for international fixtures now looming.
He added: "I can't say if I was sitting on the Ajax bench for the last time.
"I don't know more than that the clubs are speaking.