Beale has been taking criticism since his Rangers side lost the Old Firm derby at home against a weakened Celtic team just before the international break.
Rangers have found themselves trailing Celtic by four points already and they have also failed to qualify for the group stage of the Champions League, which will hurt their financial coffers.
There is already talk amongst some Rangers fans about getting rid of Beale but the club have been sticking with a manager who they backed to overhaul their squad in the summer.
However, Moore feels the Rangers boss is now walking a tightrope and in order to survive he needs to have a flawless record until Christmas and bring the Scottish League Cup back to Ibrox.
The former Rangers star said on the Go Radio Football Show: “Michael Beale is in a really tough situation at the moment.
“I believe he needs a perfect run, in terms of not dropping points, getting to Christmas and having won the League Cup final.
“Any dips or drops from here on, I can’t see any other solution than him being out of the door.”
Rangers will be desperate to get back to winning ways when they travel to St. Johnstone this weekend.